Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus \pi(A)$ and such that the union $A\cup B \subseteq X\times Y$ is still analytic?

In particular, $B$ can't be an analytic set (let along Borel) since in such case $\pi(B)$ would be analytic, whereas in some cases $X\setminus \pi(A)$ is co-analytic and not Borel, hence not analytic.