  Kang et al. provided a path realization of the crystal graph of a highest
weight module over a quantum affine algebra, as certain semi-infinite tensor
products of a single perfect crystal. In this paper, this result is generalized
to give a realization of the tensor product of several highest weight modules.
The underlying building blocks of the paths are finite tensor products of
several perfect crystals. The motivation for this work is an interpretation of
fermionic formulas, which arise from the combinatorics of Bethe Ansatz studies
of solvable lattice models, as branching functions of affine Lie algebras. It
is shown that the conditions for the tensor product theorem are satisfied for
coherent families of crystals previously studied by Kang, Kashiwara and Misra,
and the coherent family of crystals $\{B^{k,l}\}_{l\ge 1}$ of type $A_n^{(1)}$.
