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I read a sentence in Word by Word by Kory Stamper which was:

And of course we understand that "irregardless" is generally thought to be incorrect; that's why the lengthy usage paragraph suggests -- in spite of our being liberal commie descriptivists -- that people use "regardless" instead of "irregardless".

I think there should be "us" instead of "our" in the sentence. Shouldn't it be so?

Given that the author used to work as an editor for the largest American dictionary publisher, it's more likely that the phrasing is deliberate and accurate. Assuming that, we can then figure out its structure and how it is grammatical.

"Our" is a possessive pronoun, so what follows must be something that can be possessed, i.e. a noun of some kind. Consider this similar structure:

In spite of their old-fashioned opinions, they can be open to new ideas.

They possess the opinions, so we use the possessive pronoun "their". Outside of certain dialects, we would not say, "In spite of they opinions". That's not grammatical.

"Being liberal commie descriptivists" is a gerund phrase that acts like a noun, and again, something that can be possessed by us. Expanding on the previous example:

In spite of their having old-fashioned opinions, they can be open to new ideas.

Possessive + gerund phrase. It's the same with "our":

In spite of our having old-fashioned opinions, we can be open to new ideas.

Grammar-wise using "us" changes "being" from a gerund to a verb, but there's little difference in meaning. The difference is in the style, and also in the nuance. The sentence clearly refers to some earlier comment where someone called them "liberal commie descriptivists" (for insisting "irregardless" is not a word), and suggests something like:

in spite of (us possessing the quality they said we have) ...

just with fewer words. In the right context, this use of the possessive pronoun can sound a little more sophisticated.

"Our" is correct grammar here. There's a subtle difference between using us and using our, though both are correct. Using "us" implies that the "being" is a verb, so it focuses more on the act of being liberal commie descriptivists; using "our" means that that "being liberal commie descriptivists" is a gerund phrase, which function as a noun, and focuses more on that as their identity. There's hardly a difference, though.

The question is Shouldn't there be “us” instead of “our” in this sentence?

In this instance our is a pronoun, the meaning of which is belonging to us. Let's try a substitution.

in spite of our being liberal commie descriptivists

in spite of belonging to us liberal commie descriptivists

"our"
pronoun: belonging to or connected with us; the possessive form of we, used before a noun: C.E.D.