Let $X$ be a projective variety over a field $K$ of characteristic zero. Denote by $p:X_{\overline{K}} \to X$ the natural morphism, where $\overline{K}$ is the algebraic closure of $K$ and $X_{\overline{K}}:=X \times_K \overline{K}$. Let $E$ and $F$ be coherent sheaves on $E$ such that $p^*E \cong p^*F$. Does it imply that $E \cong F$?

EDIT: If necessary, assume that $E$ and $F$ are elements of the same Quot scheme on $X$.