  It was recently shown that domain walls from the spontaneous breaking of CP
symmetry are exactly stable, and must be inflated away to recover a viable
cosmology. We investigate the phenomenological implications of this result in
Nelson--Barr solutions of the strong CP problem. Combined with the upper bound
on the scale of spontaneous CP breaking necessary to suppress contributions
from dangerous, nonrenormalizable operators to $\bar{\theta}$, this puts an
upper bound on the scale of inflation and the reheating temperature after
inflation. Minimal Nelson--Barr models are therefore in tension with thermal
leptogenesis, models of large-field inflation, or potential future observations
of signals from topological remnants of an unrelated, subsequent phase
transition. We study how extending Nelson--Barr models with a new, continuous
chiral gauge symmetry can ameliorate this tension by forbidding the dangerous
dimension-five operators. In particular, we show that gauging a linear
combination of baryon number and hypercharge allows for an economic,
anomaly-free extension of the minimal Nelson--Barr model, and discuss the
phenomenological implications.
