This is an improved version of this question. Maybe it should be an edit -- I'm not sure what the MO convention is.

I'm curious, more or less, how much information one can get out of the derived category of coherent sheaves on a variety which are trivialized outside a finite set. Specifically, suppose $X$ is a smooth, proper variety of finite type over $\mathbb{C}$. Consider the category $\mathcal{C}=\text{Coh}(X)_{\text{codim }0}$ of coherent sheaves which are trivial (in the sense of being isomorphic to a power of the constant sheaf) outside a finite set of $\mathbb{C}$-points of $X$. This is a symmetric monoidal category with inner Homs. Suppose we do not know $X$, but know the category $\mathcal{C}$, together with derived data (the derived category of complexes whose cohomology is in $\mathcal{C}$, together with derived tensor products, Homs and most importantly global sections). Suppose in addition we are given the set of points of $X$ together with their formal neighborhoods, and know for every sheaf in $\mathcal{C}$ how it can be patched out of a globally trivial sheaf together with patching data at some finite set of formal neighborhoods.

How much information about $X$ does the category $\mathcal{C}$ contain? In particular can $X$ be recovered from $\mathcal{C}$?