  We investigate the effects of non-Hermiticity on interacting fermionic
systems. We do this by including non-Hermitian bilinear terms into the 3+1
dimensional Nambu--Jona-Lasinio (NJL) model. Two possible bilinear
modifications give rise to $\mathcal{PT}$ symmetric theories; this happens when
the standard NJL model is extended either by a pseudovector background field
$ig \bar\psi\gamma_5 B_\mu \gamma^\mu \psi$ or by an antisymmetric-tensor
background field $g \bar\psi F_{\mu\nu}\gamma^\mu \gamma^\nu \psi$. The three
remaining bilinears are {\it anti}-$\mathcal{PT}$-symmetric in nature, $ig
\bar\psi B_\mu \gamma^\mu \psi, ig\bar\psi \gamma_5 \psi$ and $ig\bar\psi
{1}\psi$, so that the Hamiltonian then has no overall symmetry. The
pseudovector $ig \bar\psi\gamma_5 B_\mu \gamma^\mu \psi$ and the vector $ig
\bar\psi B_\mu \gamma^\mu \psi$ combinations, are, in addition, chirally
symmetric. Thus, within this framework we are able to examine the effects that
the various combinations of non-Hermiticity, $\mathcal{PT}$ symmetry, chiral
symmetry and the two-body interactions of the NJL model have on the existence
and dynamical generation of a real effective fermion mass (a feature which is
absent in the corresponding modified massless free Dirac models) as well as on
the masses of the composite particles, the pseudoscalar and scalar mesonic
modes ($\pi$ and $\sigma$ mesons). Our findings demonstrate that $\mathcal{PT}$
symmetry is not necessary for real fermion mass solutions to exist, rather the
two-body interactions of the NJL model supersede the non-Hermitian bilinear
effects. The effects of chiral symmetry are evident most clearly in the meson
modes, the pseudoscalar of which will always be Goldstone in nature if the
system is chirally symmetric. Second solutions of the mesonic equations are
also discussed.
