  We analyze multipartite correlations in a generalized Dicke model involving
two optical modes interacting with an ensemble of two-level atoms. In
particular, we examine correlations beyond the standard bipartite entanglement
and derive exact results in the thermodynamic limit. The model presents two
superradiant phases involving the spontaneous breaking of either a
$\mathbb{Z}_2$ or $\mathrm{U}(1)$ symmetry. The latter is characterized by the
emergence of a Goldstone excitation, found to significantly affect the
correlation profiles. Focusing on the correlations between macroscopic
subsystems, we analyze both the mutual information as well as the entanglement
of formation for all possible bipartitions among the optical and matter degrees
of freedom. It is found that while each mode entangles with the atoms, the
bipartite entanglement between the modes is zero, and they share only classical
correlations and quantum discord. We also study the monogamy of multipartite
entanglement and show that there exists genuine tripartite entanglement, i.e.
quantum correlations that the atoms share with the two modes but that are not
shared with them individually, only in the vicinity of the critical lines. Our
results elucidate the intricate correlation structures underlying superradiant
phase transitions in multimode systems.
