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In Natural History magazine article, Neil DeGrasse Tyson claims the following:

The military normally knows all about the Coriolis force and thus introduces the appropriate correction to all missile trajectories. But in 1914, from the annals of embarrassing military moments, there was a World War I naval battle between the English and the Germans near the Falklands Islands off Argentina (52° south latitude). The English battle cruisers Invincible and Inflexible engaged the German war ships Gneisenau and Scharnhorst at a range of nearly ten miles. Among other gunnery problems encountered, the English forgot to reverse the direction of their Coriolis correction. Their tables had been calculated for northern hemisphere projectiles, so they missed their targets by even more than if no correction had been applied. They ultimately won the battle against the Germans with about sixty direct hits, but it was not before over a thousand missile shells had fallen in the ocean.

The physics explanation is correct - the correction due to Coriolis force is opposite on Southern hemisphere. This incident is described also in physics textbook Analytical mechanics by Hand and Finch. By direct calculation, Coriolis effect is approximately 100 meters, so this seems significant. However I'm not sure if other factors (such as wind, rocking of the boat and evasive manoeuvres of the target) would be as large.

Did this actually happen during the Battle of Falkland Islands? Was the inaccuracy due to miscalibration of artillery due to wrong Coriolis effect corrections?

Rudolph Verner, the gunnery officer of the Inflexible, which took part in the battle, wrote a short set of notes about the successes and difficulties encountered during the engagement. He writes

Deflection caused considerable difficulty, and on two occasions one gun fired some five or six consecutive rounds nearly a ship's length astern of the target.

I traced this to a turret and called for the deflection setting, which proved to be correct.

Since this error was intermittent, and always took effect in the same direction, I am convinced that it was due to the gun-layer mistaking the stem for stern, or foremast for mainmast.

The particular cruisers we were engaging made such an error particularly easy, due to the symmetrical arrangements of their masts and funnels.

Verner gives us three key insights:

• Some deflection was observed, with the worst problems occurring in a single gun.


• This deflection was not regular and continuous, as might be expected if the Coriolis force was responsible.


• The deflection, at worst, was about one ship's length.

The Inflexible measured about 160 meters in length, and the other ships were of similar size. Calculations based on the guns and naval records predict a deflection of only 82 meters, even assuming that corrections were applied for the Coriolis force but in the wrong direction, thereby making the problem worse. It seems much more likely that the bulk of the problem was not due to the Coriolis force but to poor aiming.

Unfortunately, Verner does not specify the "deflection setting" used - or even what sort of deflections it was correcting for (wind, perhaps, or the relative motion of the ships?). When he describes it as "correct", it's not clear whether that means that it is correct if the ship was in the Northern Hemisphere, or correct given that the ship was in the Southern Hemisphere. Therefore, I can't definitively rule out that the setting was incorrect. However, I can confidently say that the major deflection that the British experienced was not due primarily to the Coriolis effect.

Unless anybody produces a Royal Navy document that even does any mention of this influence of coriolis effect, this seems to be just untrue.

It seems to be an annoying urban myth. The coriolis effect is present and significant for shots at long ranges and flight duration. But it is not the only factor involved and naval gunners did not rely on tables alone. The coriolis effect for any gun on any ship at any latitude in any direction is a constant influence on accuracy.

As such systematic errors should be quickly spotted corrected for as long as the targets are still visible, that is: above the horizon if spotting is to be done from the same ship that is firing.

This myth is entirely absent from all history books I checked on for this battle. It is also missing from German accounts of the events, as evidenced by how German Wikipedia narrates the analysis:

During the battle, the two British battle cruisers fired the considerable quantity of almost 1200 305 mm grenades at the German armoured cruisers, which allows conclusions to be drawn about the training of the operators and the quality of the grenades. However, the Royal Navy did not draw any conclusions from this circumstance until after the Battle of Skagerrak one and a half years later.

A more specialised Wiki puts it this way:

Coriolis Effect
An annoying urban legend persists that the Royal Navy's shooting at the Battle of the Falklands was poor due to their equipment applying corrections for Coriolis effect in the wrong direction, as the action was in the southern hemisphere rather than the northern. The truth is, however, that no contemporary aspect of Royal Navy equipment or procedure took Coriolis effect into consideration, an extremely minor deficiency. For, even if the fable were true, if the action took place on a nearly constant bearing, and at a range that changed only slowly, even a blatant mistreatment of Coriolis effect such as its negative consideration would therefore have been a constant error, and one unlikely to be large compared to other factors affecting the proper deflection to use (such as the zig-zagging of a fleeing enemy). This fact implies that the remedy for such a miscue would have been a single spotting correction for deflection which, once made, would counteract the error for the remainder of the action.

While I think it likely that later systems of firing incorporated Coriolis corrections, a system lacking such treatment which is designed primarily to bring fire upon a maneuvering enemy is not a sad system by any means. Taken in context, Coriolis errors are a constant source of deflection error and very small in degree. The need to fire repeated salvoes which for many reasons will require spotting to put them onto the target implies that a failure to handle Coriolis effect, or even handle it entirely backwards, would not prevent a shooter from hitting his target in a prolonged engagement.

And the British Royal Navy summarises, without ever mentioning "coriolis":

System 1914

1. In the year 1914 a high standard of efficiency in the control of fire was attained; the system used was thoroughly understood and there was no lack of confidence in the ability of the fire control system to compete successfully with the accepted standards of range and general battle conditions.


2. Briefly, the rangefinder equipment was used to feed the fire control gear, and the latter was relied upon to furnish the requisite information for successful attack on a moving target. In conjunction with this, the bracket system of spotting was universally used to correct the best mean range into the actual gun range after opening fire.

The correction of the remaining factors, such as rate and deflection, was primarily dependent upon observation of fire, although great importance was attached to the use of the fire control gear as a guide.

Battle Experience

3. The earliest engagements of the war gave no cause to suspect that the firing rules were inadequate to deal with battle conditions.


4. The Battle of Heligoland Bight in 1914, fought in very low visibility was not of a character to produce any reliable evidence one way or another. It showed, however -
   The impossibility of taking ranges in low visibility conditions.


5. The action off the Falkland Islands in the same year demonstrated the following:
   
   
   • The rangefinder equipment failed to provide much information. (This was chiefly due to the range at which the action was fought, which outclassed the 9-ft rangefinders.)
   
   
   • The use of defensive tactics (zigzagging) rendered the control of fire extremely difficult, and placed a high premium on rapidity of fire as soon as the gun range was found.
   
   

USE OF 100-YARD CORRECTIONS

8. Reports were called for at the end of 1932 regarding the desirability of limiting the occasions on which corrections of 100 yards for range are permissible, in view of the frequent occasions on which range spotting corrections are insufficiently bold.


9. As a result of these reports it was not considered that a case had been made out for altering the existing rules, but it was stressed that control officers must constantly bear in mind the necessity for the use of bold spotting corrections, especially when aircraft are not available for spotting.
   _{–– ADM 186/339 C.B. 3001/1914-36: Summary
   Of Progress In Naval Gunnery, 1914-1936, Training And Staff Duties Division, Naval Staff, Admiralty, S.W., December, 1936.}

It is curious for the claim to only focus on the battle at the Falklands. As for the previous battle at Coronel, also on the Southern hemisphere, the outcome was similar. And again at Jutland, this time on the supposedly 'correct' hemisphere for correctly calibrated coriolis compensation: the British range finding and gun aiming was just inferior.

As books focusing just on that specific battle alone never mention coriolis effect, a work that analysis just the history of naval gunnery and aiming techniques concludes:

The rangefinder measured the geometric range between shooter and target, which was sometimes called the true range. Given rangefinder errors, this measurement could not be entirely accurate, but it is convenient to identify the rangefinder figure with the actual distance between ship and target. This range was not the same as the gun range, the range to which sights should be set. Gun range took into account the movement of the target while the shell was in the air and even that of the shooter while the shell was in the gun (where it shared the ship’s motion). It thus involved knowledge of how the range was changing: the range rate. The longer the range (ie, the more time the shell spent in the air), the more significant the range rate. At very long range, factors such as the rotation of the earth had to be taken into account. It began to matter that a ship was able to measure her own speed. That was difficult: only in about 1912 did the Royal Navy obtain an electric log (measuring speed). Other navies were probably in about the same position: the Germans license-produced the British log.

The British were fairly sure, moreover, that their understanding of gunnery was far in advance of any other navy: in 1906 DNO’s assistant Captain Harding remarked that foreign navies did not yet understand the difference between geometric and gun range.26 Only recently had British officers realised how important it was to know the geometric range precisely, rather than depend on spotting beginning with an approximate range. Presumably this referred partly to Captain W C Pakenham’s comments during the Russo-Japanese War (Pakenham was Royal Naval attaché to Japan at the time): ‘Outside the Service the impossibility of continuous use of the rangefinder and therefore the importance of a knowledge of the rate of change [range rate] is not recognised, consequently the means of its determination are unsought for.’ No one had tried to make a rangefinder record its output automatically, and no one (apart from Pollen, see chapter 2) had realised the importance of using a gyro to eliminate yaw from rangefinder bearing readings. The Germans were probably the most advanced foreign navy at this time. Little was known of their thinking, but the evidence of what they were using (sextants with a few unmodified Barr & Stroud rangefinders) and of articles in their main annual publication, Nauticus, suggested that they were not working along British lines.

The range rate

Successful gunnery required that the position of the target be projected ahead, ultimately to the moment at which a shell might be expected to hit. To do that, the shooter had to calculate the rates at which the range and bearing of the target changed; they were usually called the range and bearing rates. Calculation was difficult because neither was constant, and because each depended on the other. Alternatively, one might think in terms of the vector (magnitude and direction) pointing from shooter to target. The change in this vector was another vector which might be called the rate vector. It could be expressed as two components, one along the line of fire and one across it. The rate along was usually called the range rate. The rate across was usually called deflection. Its magnitude was the bearing rate multiplied by the range.

The Falklands

In December 1914 two British battlecruisers fought Admiral Graf von Spee’s Pacific Squadron, which had recently sunk the HMS Good Hope at Coronel.13 This time visibility was excellent (only in the last hour of the battle did it fall to 15,000 yards), and both squadrons steamed at high speed. Neither British ship had a functioning director or a Dreyer Table. Both ships found their fire control hampered by funnel smoke, so that although her fore conning tower and A turret never lost sight of the enemy, in Invincible the fore top occasionally lost sight, and P, Q, and X turrets were much affected. Rangefinding was very difficult due to the long range, funnel smoke, splashes and spray from the enemy. Rate-keeping was difficult at best, due to the enemy’s zigzagging as well as to the very long range (variations in range were almost undetectable). Gunners found it difficult to stay on a point of aim, sometimes mistaking the target’s bow for her stern. On the other hand, according to prisoners, British shells performed well, penetrating and exploding deep in the ships. Even so, Gneisenau took fifty 12in hits before sinking. It was no great surprise that the British ships used up most of their ammunition: one 12in gun in Inflexible fired 109 rounds (the ship was designed to carry eighty for that gun).

To the surprise of the British, von Spee’s ships zigzagged to avoid being hit, even thought that made hits by their own guns unlikely. As crack gunnery ships, the Germans were expected to fire at maximum range, but the actual figure for their 8.2in guns, 16,000 yards, seems to have surprised the British. The Germans straddled (without hitting) at 15,000 yards. The Germans persistently fired salvoes (the British thought, wrongly, that they were using directors), and their direction and fire discipline were excellent. The British were impressed by the effect of plunging shells at such ranges, and by the blast effect of the German fire. The German survivors stressed, and the British noted, that slow British fire made it easier for their own gunlayers. It also made spotting easier, because the British ships were much less completely enveloped in the smoke of their own guns. This may have been the first of many British observations that their firing techniques were far too deliberate.
_{-- Norman Friedman: "Naval Firepower. Battleship Guns And Gunnery In The Dreadnought Era", Seaforth Publishing: Barnsley, 2008.}