With a slight modification of the original definition by Billera and Sturmfels, it is shown that the theory of fibre polytopes extends to one of mixed fibre polytopes. Indeed, there is a natural surjective homomorphism from the space of tensor weights on polytopes in a euclidean space V to the corresponding space of such weights on fibre polytopes in a subspace of V. Moreover, these homomorphisms compose in the correct way; this is in contrast to the original situation of the fibre polytope construction.