For any essentially small, rigid and idempotent-complete tensor triangulated (TT for short) category $\mathcal{T}$ Balmer (The spectrum of prime ideals in tensor triangulated categories) constructs a locally ringed topological space Spec($\mathcal{T}$).

When $\mathcal{T}$ is the derived category of perfect complexes, $D^o(S)$, over a quasi-compact quasi-separated (qcqs) scheme $S$, the construction recovers $S$ as a locally ringed space together with its structure sheaf from the tensor triangular structure of $D^o(S)$.

My questions are:

(1) Given a TT category, can we tell when it is equivalent to $D^o(X)$ for a qcqs scheme $X$?

Let's call the TT categories that fulfill the criterion *schematic*.

(2) If we take an essentially small additive category and put different schematic tensor triangulated structures on it, we'd get different schemes via Balmer's construction. Can all flat families of schemes of a certain type (e.g., smooth curves of genus $g$ over a field $k$) be obtained by varying the tensor structure alone?

(3) What are some natural discrete invariants of TT-categories and how can we build their moduli spaces after fixing values of the discrete invariants?