  Symmetries associated with complex conjugation and Hermitian conjugation,
such as time-reversal symmetry and pseudo-Hermiticity, have great impact on
eigenvalue spectra of non-Hermitian random matrices. Here, we show that
time-reversal symmetry and pseudo-Hermiticity lead to universal level
statistics of non-Hermitian random matrices on and around the real axis. From
the extensive numerical calculations of large random matrices, we obtain the
five universal level-spacing and level-spacing-ratio distributions of real
eigenvalues, each of which is unique to the symmetry class. Furthermore, we
analyse spacings of real eigenvalues in physical models, such as bosonic
many-body systems and free fermionic systems with disorder and dissipation. We
clarify that the level spacings in ergodic (metallic) phases are described by
the universal distributions of non-Hermitian random matrices in the same
symmetry classes, while the level spacings in many-body localized and Anderson
localized phases show the Poisson statistics. We also find that the number of
real eigenvalues shows distinct scalings in the ergodic and localized phases in
these symmetry classes. These results serve as effective tools for detecting
quantum chaos, many-body localization, and real-complex transitions in
non-Hermitian systems with symmetries.
