  Nonreciprocal interactions are commonplace in continuum-level descriptions of
both biological and synthetic active matter, yet studies addressing their
implications for time-reversibility have so far been limited to microscopic
models. Here, we derive a general expression for the average rate of
informational entropy production in the most generic mixture of conserved phase
fields with nonreciprocal couplings and additive conservative noise. For the
particular case of a binary system with Cahn-Hilliard dynamics augmented by
nonreciprocal cross-diffusion terms, we observe a non-trivial scaling of the
entropy production rate across a parity-time symmetry breaking phase
transition. We derive a closed-form analytic expression in the weak-noise
regime for the entropy production rate due to the emergence of a macroscopic
dynamic phase, showing it can be written in terms of the global polar order
parameter, a measure of parity-time symmetry breaking.
