It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one can find explicit counterexamples. However, they do satisfy a weak form of choice called WISC, and do so even when only ZF + WISC is chosen as the base theory.

The question is: are any other choice principles inherited by Grothendieck topoi like this? The question may also include axioms that are mutually incompatible with the axiom of choice but still imply taking a position on the matter while being fairly natural, such as for example the axiom of determinacy.