Finite unramified analytic coverings vs finite etale coverings

(Using the notation from the question) $\mathcal F$ is a coherent sheaf of $\mathcal O_{\overline X}$-algebras. Then $Z={\rm Spec}_{\overline X}\, \mathcal F\to \overline{X}$ is a finite morphism between projective schemes. Looking at the construction of ${\rm Spec}_{\overline X} \mathcal F$ should tell you that $\overline Y\simeq Z^{\rm an}$.