Let $X$ be a topological space. For every point $x \in X$ let $R_x$ be a local ring. Under what (necessary / sufficient / necessary and sufficient) conditions is there a sheaf ${\cal O}_X$ such that $(X,\mathcal{O}_X)$ is a locally ringed space with $\mathcal{O}_{X,x} \cong R_x$ for all $x \in X$?

If $X$ is discrete, we need no conditions, and we can take the sheaf $\mathcal{O}_X(U)=\prod_{x \in U} R_x$. But in general, a necessary condition is that $x \prec y$ gives a ring homomorphism $R_x \to R_y$, and that these are compatible in the sense that $x \mapsto R_x$ extends to a functor from the specialization preorder of $X$ to the category of rings. We will also have $\mathcal{O}_X(U) \subseteq \varprojlim_{x \in U} R_x$, but probably no equality.