  Many-body localization is shown to suppress imaginary parts of complex
eigenenergies for general non-Hermitian Hamiltonians having time-reversal
symmetry. We demonstrate that a real-complex transition, which we conjecture
occurs upon many-body localization, profoundly affects the dynamical stability
of non-Hermitian interacting systems with asymmetric hopping that respect
time-reversal symmetry. Moreover, the real-complex transition is shown to be
absent in non-Hermitian many-body systems with gain and/or loss that breaks
time-reversal symmetry, even though the many-body localization transition still
persists.
