  We consider spin systems between a finite number $N$ of "species" or "phases"
partitioning a cubic lattice $\mathbb{Z}^d$. We suppose that interactions
between points of the same phase are coercive, while between point of different
phases (or, possibly, between points of an additional "weak phase") are of
lower order. Following a discrete-to-continuum approach we characterize the
limit as a continuum energy defined on $N$-tuples of sets (corresponding to the
$N$ strong phases) composed of a surface part, taking into account
homogenization at the interface of each strong phase, and a bulk part which
describes the combined effect of lower-order terms, weak interactions between
phases, and possible oscillations in the weak phase.
