This is motivated by a previous question of mine, but I think it is ultimately more interesting (and hopefully easier to answer in the positive). In that question, a class of games *(on $\omega$, of length $\omega$)* is considered where the payoff set for such a game is determined by $\omega_1$-many, rather than the usual $2^\omega$-many, yes/no facts. Call such a class of games **overdetermined**; I'm interested in whether there are, provably in ZFC+$\neg$CH, undetermined overdetermined games (heh).

Formally, say that a game with payoff set $A$ is **$\Gamma$-overdetermined** (for $\Gamma$ a pointclass) if $A$ is $E$-invariant for some equivalence relation $E\in\Gamma$ with $\omega_1$-many classes.
Obviously this is a silly notion in full generality: take $E=\{(a, b): a\in A\iff b\in A\}$. Things become interesting, however, when we restrict attention to reasonably-definable $E$. For example, every copy/diagonalize game on ordinals (see the question linked above) is overdetermined with respect to a (fixed) $(\Sigma^1_1\vee\Pi^1_1)$-relation: changing notation slightly, a play $\pi$ yields a sequence of linear orderings $L_i^\pi$ $(i\in\omega)$, and two plays $\pi,\chi$ are guaranteed to be won by the same player if $L_i^\pi\approx L_i^\chi$ for each $i\in\omega$, where $A\approx B$ for linear orders $A, B$ if $A$ and $B$ are isomorphic or each is ill-founded. And this equivalence relation has only $\omega_1$-many classes.

My question is:

Working in ZFC + $\neg$CH, can we prove the existence of an undetermined game which is overdetermined by some "tame" pointclass (e.g. projective, or even better $\Sigma^1_1\vee\Pi^1_1$)?

I strongly suspect that the answer is "yes," even for $\Sigma^1_1\vee\Pi^1_1$, but I don't immediately see how to prove it. Note that I'm allowing the game itself to be as horrible as desired, as long as it respects some nice equivalence relation with few classes (incidentally, I'd also be happy to replace "$\omega_1$" with "$<2^\omega$") - the point is to restrict the usefulness of choice in terms of building an undetermined game by ensuring that any naive construction via transfinite recursion has to "end too early" to be guaranteed to work.

Beyond this question, I'm interested in any literature on games of arbitrary complexity but which respect some "tame" equivalence relation with few classes; I haven't managed to find any on my own.