Simon Thomas already mentioned that a winning strategy can be recursively coded into a real. I thought it might be a good idea to write down one such coding explicitly:

Let $A \subseteq \mathcal{N}$ and let $\Sigma$ be a winning strategy for $G_{A}$ (say for player $II$, the other case is virtually the same). Then
$$
\Sigma \colon \bigcup_{n < \omega} \mathbb{N}^{n+1} \to \mathbb N.
$$
Let $\mathbb P = \{ p_i \mid i < \omega \}$ be the increasing enumeration of all primes and let
$$
\Sigma^c\colon \mathbb N \to \mathbb N, \prod_{p \in \mathbb P} p^{n_p} \mapsto \Sigma(n_3, \ldots, n_{p_{(n_2 + 1)}}).
$$
Then $\Sigma^c \in \mathcal{N}$ is a recursive coding of $\Sigma$.