  We introduce the operation of forming the tensor product in the theory of
analytic Frobenius manifolds. Building on the results for formal Frobenius
manifolds which we extend to the additional structures of Euler fields and flat
identities, we prove that the tensor product of pointed germs of Frobenius
manifolds exists. Furthermore, we define the notion of a tensor product diagram
of Frobenius manifolds with factorizable flat identity and prove the existence
such a diagram and hence a tensor product Frobenius manifold. These diagrams
and manifolds are unique up to equivalence. Finally, we derive the special
initial conditions for a tensor product of semi--simple Frobenius manifolds in
terms of the special initial conditions of the factors.
