Subgroups of direct products of elementarily free groups

We exploit Zlil Sela’s description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if $G_1,\mathrm{.}\mathrm{.}\mathrm{.},G_n$ are in E then a subgroup $\Gamma\subset G_1\times…\times G_n$isoftype$\FP_n$ifandonlyif$\Gamma$ is itself, up to finite index, the direct product of at most $n$ groups from $\mathcal{E}$. This answers a question of Sela.