Let G1, G2 be real reductive groups and (π, V ) be a smooth admissible representation of G1×G2 . We prove that (π, V ) is irreducible if and only if it is the completed tensor product of (πi, Vi), i = 1, 2, where (πi, Vi) is a smooth, irreducible, admissible representation of moderate growth of Gi , i = 1, 2. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG09, Appendix A] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair H ⊂ G of real reductive groups is equivalent to the usual Gelfand property of the pair ∆H ⊂ G×H . Mathematics Subject Classification 2000: 20G05, 22D12, 22E47.