Summary: Let \(X\) be a smooth projective variety over \(\mathbb{C}\) with big (anti)canonical bundle. It is known that for such \(X\) the Balmer spectrum of the tensor triangulated category of perfect complexes \(\operatorname{Perf}(X)\), equipped with the derived tensor product \(\otimes_X^{\mathbb{L}} \), recovers the space \(X\). We study the possible tensor triangulated category structures one can put on \(\operatorname{Perf}(X)\). As an application, we prove a monoidal version of the well-known Bondal-Orlov reconstruction theorem.