  In this article, we investigate a unique prime factorization property for
infinite tensor product factors. We provide several examples of type II and III
factors which satisfy this property, including all free product factors with
diffuse free product components. In the type III setting, this is the first
classification result for infinite tensor product non-amenable factors. Our
proof is based on Popa's intertwining techniques and a characterization of
relative amenability on the continuous cores.
