  We investigate the photon emission properties of the driven-dissipative Rabi
model in the so-called ultrastrong and deep strong coupling regimes, where the
atom-cavity coupling rate $g$ becomes comparable or larger than the cavity
frequency $\omega_c$. By solving numerically the master equation in the
dressed-state basis, we compute the output field correlation functions in the
steady-state for a wide range of coupling rates. We find that, as the
atom-cavity coupling strength increases, the system undergoes multiple
transitions in the photon statistics. In particular, a first sharp
anti-bunching to bunching transition, occurring at $g\sim0.45\omega_c$, leading
to the breakdown of the photon blockade due to the counter-rotating terms, is
shown to be the consequence of a parity shift in the energy spectrum. A
subsequent revival of the photon blockade and the emergence of the
quasi-coherent statistics, for even larger coupling rates, are attributed to an
interplay between the nonlinearity in the energy spectrum and the transition
rates between the dressed states.
