Hyland's paper “Variations on realizability: realizing the propositional axiom of choice”, Math. Struct. in Comp. Science12 (2002) 295–317, which tries to construct various models of constructive math satisfying at least some form of Choice (the “propositional” axiom of Choice $∀x∈X. ∃y∈Y. φ(x,y) \Rightarrow ∃f∈Y^X. ∀x∈X. φ(x,f(x))$) seems very relevant to this question. Hyland attributes the question to Maietti and tries to answer it at least partially. (But I'm not sure if it's precisely the same as this post's question or just related.)
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