  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 $G_1,...,G_n$ are in E then a subgroup $\Gamma\subset
G_1\times...\times G_n$ is of type $\FP_n$ if and only if $\Gamma$ is itself,
up to finite index, the direct product of at most $n$ groups from $\mathcal E$.
This answers a question of Sela.
