Given two finite groups (or compact Lie groups in general) \(G_1\) and \(G_2\), it is well-known that \(\pi_1\otimes\pi_2\) is an irreducible complex linear representation of \(G_1\times G_2\) for any irreducible complex linear representation \(\pi_1\) (and \(\pi_2\)) of \(G_1\) (and \(G_2\)). Conversely, any irreducible complex linear representation of \(G_1\times G_2\) is of this form. For smooth admissible representations of \(p\)-adic groups, a similar statement was proved by Bernstein-Zelevinsky and also by Flath. In this paper, the authors prove a similar statement for smooth admissible Fréchet representations of moderate growth of real reductive Lie groups. The proof uses the equivalence between the category of such representations and the category of Harish-Chandra \((\mathfrak{g},K)\)-modules. As a corollary they show that for reductive groups \(H\subset G\), \((G,H)\) is a multiplicity-free pair if and only if \((G\times H,\Delta(H))\) is a Gelfand pair.