  We study strongly graded vertex algebras and their strongly graded modules,
which are conformal vertex algebras and their modules with a second, compatible
grading by an abelian group satisfying certain grading restriction conditions.
We consider a tensor product of strongly graded vertex algebras and its tensor
product strongly graded modules. We prove that a tensor product of strongly
graded irreducible modules for a tensor product of strongly graded vertex
algebras is irreducible, and that such irreducible modules, up to equivalence,
exhaust certain naturally defined strongly graded irreducible modules for a
tensor product of strongly graded vertex algebras. We also prove that certain
naturally defined strongly graded modules for the tensor product strongly
graded vertex algebra are completely reducible if and only if every strongly
graded module for each of the tensor product factors is completely reducible.
These results generalize the corresponding known results for vertex operator
algebras and their modules.
