Suppose $M$ is a countable transitive model of $\mathsf{ZFC}$ (maybe more). Suppose $x, y \in {}^\omega\omega \cap M$ and $M \models y = x^\sharp$. Is it true (possible with additional assumptions), that $V \models y = x^\sharp$?

From Kanomori's book, $y = x^\sharp$ is $\Pi_2^1$. However, I can not apply Shoenfield absoluteness since $M$ is countable and does not have the countable ordinals of $V$.

An even stronger general question may be: are there conditions on $M$ or $V$ such that $\Pi_2^1$ absoluteness can hold between the universe and the countable model $M$.

Thanks for any insight on these questions.