Linear preservers of tensor product of unitary orbits, and product numerical range

It is shown that the linear group of automorphism of Hermitian matrices which preserves the tensor product of unitary orbits is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. The result is then applied to show that automorphisms of the product numerical ranges have the same structure.