Suppose $X$ is a nice (finite type over $\mathbb{C}$, smooth and proper if necessary) variety. Suppose we're given the $\mathbb{C}$-points of $X$ as a set and for any finite subset $S\subset X$ we know the DG algebra of derived sections $\Gamma_{DG}(O_X, X\setminus S)$ of the sheaf of functions, together with the transition maps when we replace $S$ by $S'\supset S$. Can we reconstruct $X$?

Edit: I think the answer to this question is "no", and here is a better version of the question: How much of a variety can be reconstructed from codimension-zero data?