  In the present study, the interplay among interaction, topology,
quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on
a one-dimensional lattice with asymmetry hoppings and quasiperiodic onsite
potentials is selected. This model, which preserves time-reversal symmetry
(TRS), will exhibit three types of phase transition: real-complex transition of
eigenenergies, topological phase transition and many-body localization (MBL)
phase transition. For thereal-complex transition, it is found that the
imaginary parts of the eigenenergies are always suppressed by the MBL.
Moreover, by calculating the winding number, a topological phase transition can
be revealed with the increase of potential amplitude, and we find that the
behavior is quite different from the single-particle systems. Based on our
numerical results, we conjecture that these three types of phase transition
occur at the same point in the thermodynamic limit, and the MBL transition of
quasiperiodic system and disordered system should belong to different
universality classes. Finally, we demonstrate that these phase transitions can
profoundly affect the dynamics of the non-Hermitian many-body system.
