On the following page, Taranovsky is talking about his "Determinacy Maximum" axiom: http://web.mit.edu/dmytro/www/DeterminacyMaximum.htm
He also justifies the choice of the name, by pointing out that many simple extensions of the theorem lead to inconsistency.
One of the results is theorem 3, giving an example of a game which has length $\omega_1+\omega$ and ordinal definable payoff and is undetermined. However, the proof uses dichotomy "$\omega_1=\Bbb R$ or $\omega_1<\Bbb R$" which need not hold when not assuming AC (e.g. it fails if we assume AD).
My question thus is
Can Taranovsky's game be modified so that it is undetermined without assuming axioms other than ZF?
Thanks in advance.
(PS. Is game-theory tag fitting for this kind of question? If not, what tag is to be used in the future?)
Added: I have noticed that Wikipedia claims existence of such long undetermined game on Determinacy article, but it has no references nor claim about background system.