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I am an intermediate student of Russian, and in one of my courses I came across the following example:

Тут два белых стула. — Two white chairs are here.

My understanding is that the genitive is generally used with numbers, and here the adjective белый is declined to the genitive plural белых, so I would expect the noun to also be declined to the genitive plural стульев. However, the example uses the genitive singular стула for "chairs".

Why is this the case, and is there a "rule" that addresses this type of sentence?

For clarity and contrast, here is another example in the genitive that declines the way I was expecting:

У меня нет двух белых стульев. — I do not have two white chairs.

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This is a complex area of Russian grammar. All grammar texts address this; but Derrek Offord's "Modern Russian: An Advanced Grammar Course" [Offord][1] addresses it better than many, at least for non-native learners. It's out of print now, but it can still be found. On page 222, here's what Offord has to say about this very topic:

When два/две, оба/обе, полтора/полторы, три & четыре are themselves are used in the nominative or accusative case the noun that follows them is in the genitive singular case.

e.g. два стула

But why genitive plural for the adjective белых in your first example? Again, Offord p.222:

If an adjective is used with a noun after any of these numerals then it will be in the genitive plural if the noun is masculine or neuter, and in the nominative/accusative plural (or possibly genitive plural) if the noun is feminine.

Therefore, Тут два белых стула, but Тут две белые книги.

The cognitive hurdle here is created by the breakage of the seemingly inviolable rule of adjective-noun agreement in case and number.

[1]: Offord, Derek. Modern Russian: An Advanced Grammar Course. Reprinted with corr, Bristol Classical Press, 1994.

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