  We investigate the impact of non-Hermiticity on the thermodynamic properties
of interacting fermions by examining bilinear extensions to the $3+1$
dimensional $SU(2)$-symmetric Nambu--Jona-Lasinio (NJL) model of quantum
chromodynamics at finite temperature and chemical potential. The system is
modified through the anti-$PT$-symmetric pseudoscalar bilinear
$\bar{\psi}\gamma_5 \psi$ and the $PT$-symmetric pseudovector bilinear $iB_\nu
\,\bar{\psi}\gamma_5\gamma^\nu \psi$, introduced with a coupling $g$. Beyond
the possibility of dynamical fermion mass generation at finite temperature and
chemical potential, our findings establish model-dependent changes in the
position of the chiral phase transition and the critical end-point. These are
tunable with respect to $g$ in the former case, and both $g$ and $|B|/B_0$ in
the latter case, for both lightlike and spacelike fields. Moreover, the
behavior of the quark number, entropy, pressure, and energy densities signal a
potential fermion or antifermion excess compared to the standard NJL model, due
to the pseudoscalar and pseudovector extension respectively. In both cases
regions with negative interaction measure $I = \epsilon-3p$ are found. Future
indications of such behaviors in strongly interacting fermion systems, for
example in the context of neutron star physics, may point toward the presence
of non-Hermitian contributions. These trends provide a first indication of
curious potential mechanisms for producing non-Hermitian baryon asymmetry. In
addition, the formalism described in this study is expected to apply more
generally to other Hamiltonians with four-fermion interactions and thus the
effects of the non-Hermitian bilinears are likely to be generic.
