Let $(R_i)_{i \in I}$ be a family of rings and suppose that we have compatible isomorphisms of subschemes of the spectra of the $R_i$ so that we are able to glue the corresponding affine schemes $\text{Spec}(R_i)$.
Question: Are there sufficient condition on the rings $R_i$ and on the given maps such that the glued scheme is again affine.
I have the feeling that this works out in very special situations and this is also my motivation for the question: Let $A$ be an integral domain with quotient field $K$. The space $\text{Zar}(K \mid A)$ is the set of all valuation overrings of $A$ endowed with the Zariski topology. It is known that $\text{Zar}(K\mid A)$ is a spectral space, that is, it is homeomorphic to the spectrum of a ring. Indeed, there are explicit constructions of Kronecker function rings, which are Bézout domains and have spectrum homeomorphic to $\text{Zar}(K\mid A)$ for given $A$. But these constructions are quite out-of-the-hat, in my opinion, so here is my idea to this problem:
The subspace $\text{Zar}(K\mid V)$ is totally ordered by set-theoretical inclusion and homeomorphic to $\text{Spec}(V)$ by localization, for each $V \in \text{Zar}(K\mid A)$. Moreover, $\text{Zar}(K\mid A)$ is covered by those. So if we glue them at their overlapping subschemes, we get a scheme whose underlying space is homeomorphic to $\text{Zar}(K\mid A)$. If this scheme would be affine, then we have reached the goal.
Thank you for your help!