  Real-world systems are characterized by complex interactions of their
internal degrees of freedom, while living in ever-changing environments whose
net effect is to act as additional couplings. Here, we introduce a paradigmatic
interacting model in a switching, but unobserved, environment. We show that the
limiting properties of the mutual information of the system allow for a
disentangling of these two sources of couplings. Further, our approach might
stand as a general method to discriminate complex internal interactions from
equally complex changing environments.
