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 $A_n^{(1)}$.