  We study the nonreciprocal Cahn-Hilliard model with thermal noise as a
prototypical example of a generic class of non-Hermitian stochastic field
theories, analyzed in two companion papers [Suchanek, Kroy, Loos,
ArXiv:2303.16701 (2023); Suchanek, Kroy, Loos, ArXiv:2305.05633 (2023)]. Due to
the nonreciprocal coupling between two field components, the model is
inherently out of equilibrium and can be regarded as an active field theory.
Beyond the conventional homogeneous and static-demixed phases, it exhibits a
traveling-wave phase, which can be entered via either an oscillatory
instability or a critical exceptional point. By means of a Fourier
decomposition of the entropy production rate, we quantify the associated
scale-resolved time-reversal symmetry breaking, in all phases and across the
transitions, in the low-noise regime. Our perturbative calculation reveals its
dependence on the strength of the nonreciprocal coupling. Surging entropy
production near the static-dynamic transitions can be attributed to
entropy-generating fluctuations in the longest wavelength mode and heralds the
emerging traveling wave. Its translational dynamics can be mapped on the
dissipative ballistic motion of an active (quasi)particle.
