  A simple polytope $P$ is called $B$-rigid if its combinatorial type is
determined by the cohomology ring of the moment-angle manifold $\mathcal{Z}_P$
over $P$. We show that any tensor product decomposition of this cohomology ring
is geometrically realized by a product decomposition of the moment-angle
manifold up to equivariant diffeomorphism. As an application, we find that
$B$-rigid polytopes are closed under products, generalizing some recent results
in the toric topology literature. Algebraically, our proof establishes that the
Koszul homology of a Gorenstein Stanley-Reisner ring admits a nontrivial tensor
product decomposition if and only if the underlying simplicial complex
decomposes as a join of full subcomplexes.
