  We explore the physical consequences of a scenario when the standard
Hermitian Nambu--Jona-Lasinio (NJL) model spontaneously develops a
non-Hermitian PT-symmetric ground state via dynamical generation of an
anti-Hermitian Yukawa coupling. We demonstrate the emergence of a noncompact
non-Hermitian (NH) symmetry group which characterizes the NH ground state. We
show that the NH group is spontaneously broken both in weak- and
strong-coupling regimes. In the chiral limit at strong coupling, the NH ground
state develops inhomogeneity, which breaks the translational symmetry. At weak
coupling, the NH ground state is a spatially uniform state, which lies at the
boundary between the PT-symmetric and PT-broken phases. Outside the chiral
limit, the minimal NJL model does not possess a stable non-Hermitian ground
state.
