  In this paper we study intersection configurations -- which describe the
behaviour of multiple (finite) intersections of subgroups with respect to
finite generability -- in the realm of free and free times free-abelian (FTFA)
groups. We say that a configuration is realizable in a group $G$ if there exist
subgroups $H_1,\ldots , H_k \leqslant G$ realizing it.
  It is well known that free groups $\mathbb{F}_n$ satisfy the Howson property:
the intersection of any two finitely generated subgroups is again finitely
generated. We show that the Howson property is indeed the only obstruction for
multiple intersection configurations to be realizable within nonabelian free
groups.
  On the contrary, FTFA groups $\mathbb{F}_n \times \mathbb{Z}^m$ are well
known to be non-Howson. We also study multiple intersections within FTFA
groups, providing an algorithm to decide, given $k\geq 2$ finitely generated
subgroups, whether their intersection is again finitely generated and, in the
affirmative case, compute a `basis' for it. We finally prove that any
intersection configuration is realizable in a FTFA group $\mathbb{F}_n \times
\mathbb{Z}^m$, for $n\geq 2$ and big enough $m$. As a consequence, we exhibit
finitely presented groups where every intersection configuration is realizable.
