  We study fluctuating field models with spontaneously emerging dynamical
phases. We consider two typical transition scenarios associated with
parity-time symmetry breaking: oscillatory instabilities and critical
exceptional points. An analytical investigation of the low-noise regime reveals
a drastic increase of the mesoscopic entropy production toward the transitions.
For an illustrative model of two nonreciprocally coupled Cahn-Hilliard fields,
we find physical interpretations in terms of actively propelled interfaces and
a coupling of modes near the critical exceptional point.
