A simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold ZP
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.