Given a set $A$, Borel determinacy for $A$ is the theorem (of $\mathsf{ZFC}$) asserting that every Borel subset of $A^\omega$ is determined. That is, if I and II take turns playing members of $A$, and the elements played form a sequence $x = (x_0,x_1,\dots)$, then for all Borel $X \subseteq A^\omega$ either I has a strategy to ensure that $x \in X$ or II has a strategy to ensure that $x \in X^c$.
It's known that "Closed determinacy for all $A$" is equivalent to the axiom of choice (over $\mathsf{ZF}$). How much choice is needed for "Borel determinacy for all Polish space $A$"?