  We study a generalization of a recently introduced Dicke trimer model [Phys.
Rev. Lett. 128, 163601, Phys. Rev. Research 5, L042016], which allows for
cavity losses and unbalanced light-matter interactions (in which rotating and
counter-rotating terms can be tuned independently). We find that in the extreme
unbalanced limit, the $U(1)$ symmetry of the Tavis-Cummings model is restored,
qualitatively altering the critical phenomena in the superradiant phase due to
the presence of a zero-energy mode. To analyze this general regime, we develop
a semiclassical theory based on a re-quantization technique. This theory also
provides further physical insight on a recently reported anomalous finite
critical fluctuations in the time-reversal broken regime. Moving to the
open-Dicke case, by introducing local dissipation to the cavities, we observe
the emergence of a rich range of nonequilibrium phases characterized by trivial
and non-trivial dynamical signatures. In the former case, we identify, when
time-reversal symmetry is present, a new stationary phase that features
superradiant states in two of the three cavities and a normal state in the
other cavity. In the latter case, we observe the emergence of dynamical phases
in which the system exhibits superradiant oscillations, characterized by
periodic or chaotic phase space patterns. The landscape of transitions
associated with these dynamical phases features a wide range of qualitatively
different behaviours such as Hopf bifurcations, anomalous Hopf bifurcations,
collisions between basins of attraction, and exterior crises. We highlight how
the two-critical-scalings feature of the closed model is robust under
dissipation while the phenomenon of anomalous finite critical fluctuations
becomes a mean-field scaling in the open model.
