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