For the purpose of this post, I will say that *the* Gale-Stewart game is the infinite two-player game of perfect information where players I and II alternate playing natural numbers, with I going first. A round of the game looks like: I plays $n_0$, II plays $n_1$, I plays $n_2$, II plays $n_3$, etc. The outcome of this round is the sequence $(n_i)_{i\in\omega}\in\omega^\omega$.

Given a set $A\subseteq\omega^\omega$, I will speak of a player *having a strategy for playing into $A$*, rather than using the usual winning/losing terminology. If $\sigma$ is a strategy for one of the players, I denote by $[\sigma]\subseteq\omega^\omega$ the set of all outcomes of the game when that player follows $\sigma$.

**Question:** Suppose we are given a set $C\subseteq\omega^\omega\times\omega^\omega$. Under which circumstances do there exists strategies $\sigma$ and $\tau$ for players (either one) in the Gale-Stewart game such that one of $[\sigma]\times[\tau]\subseteq C$ or $([\sigma]\times[\tau])\cap C=\emptyset$ holds?

Is it sufficient that $C$ is determined? To put that another way, is there a game which encodes this property? Or is there a (reasonably definable) counterexample?

The first thing that comes to mind is the game where the two players alternate, with I going first, playing pairs of natural numbers $(n_i,m_i)$, and whose outcome is the pair $((n_i)_{i\in\omega},(m_i)_{i\in\omega})\in\omega^\omega\times\omega^\omega$. If $C$ is determined, then either I has a strategy for playing into $C$ or II has a strategy for playing into its complement in this game. Suppose player I has a strategy for playing into $C$. From this, it is easy to construct two strategies $\sigma$ and $\sigma'$ in the Gale-Stewart game for player I with $[\sigma]\subseteq\pi_0(C)$ and $[\sigma']\subseteq\pi_1(C)$, where $\pi_0$ and $\pi_1$ are the first and second coordinate projections, respectively: read off the first (or second) coordinate of I's play in the game with pairs while II plays their move in the Gale-Stewart in the first (or second) coordinate, and an arbitrary number in the other coordinate. However, I have no reason to suspect that $[\sigma]\times[\sigma']\subseteq C$. One issue seems to be a lack of independence in the coordinates played according to a strategy; each of played coordinates can depend on either of the previously played coordinates.

**Edit:** Joel's answer below precludes the possibility that the strategies are from the *same* player in the Gale-Stewart game, but I want to also address the possibility that alternate players have such strategies, e.g., I has a strategy $\sigma$ and II has a strategy $\tau$ such that $[\sigma]\times[\tau]\subseteq C$ or $([\sigma]\times[\tau])\cap C=\emptyset$.