Are all supergravity theories Yang-Mills squared?

Using simple symmetry arguments we classify the ungauged D=4, N=2 supergravity theories, coupled to both vector and hyper multiplets through homogeneous scalar manifolds, that can be built as the product of N=2 and N=0 matter-coupled Yang-Mills gauge theories. This includes all such supergravities with two isolated exceptions: pure supergravity and the T3 model.