Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X_L$ for a finite extension $L$ of $K$. This gives a cocycle condition on a quasi-coherent sheaf on $X_L$ to descend to $X$ (see for example page 139, section 6.2 of "Neron Models" by S. Bosch and others).

I am looking for an analogous result for the absolute Galois group i.e., if we take a coherent sheaf $E$ on $X_{\bar{K}}$ which satisfies the analogous cocycle condition for any pair of elements $\sigma, \tau \in \mathrm{Gal}(\bar{K}/K)$, does there exist a (quasi)-coherent sheaf $F$ on $X$ such that $F \otimes_K \bar{K} \cong E$?

Any hint/reference on this topic will be most helpful.

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