(using 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 $\overline Y\simeq Z^{\rm an}$.
How about this: (using the notation from the question) $\mathcal F$ is a coherent sheaf of $\mathcal O_{\overline X}$-algebras. Then $Z={\rm Spec}_{\overline X}\, \mathcal F\to X$\overline{X}$ is a finite morphism between projective schemes. Looking at the construction of${\rm Spec}_{\overline X}\, \mathcal F$should tell you that$\overline Y\simeq Z^{\rm an}$. 1 How about this: (using the notation from the question)$\mathcal F$is a coherent sheaf of$\mathcal O_{\overline X}$-algebras. Then $Z={\rm Spec}_{\overline X}\, \mathcal F\to X$ 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}\$.