  We develop a generalized Aufbau principle for non-Hermitian systems that
allows for building up the configurations of indistinguishable particles. The
Aufbau rule of non-Hermitian systems is unexpectedly shown to be identical to
that developed in Hermitian systems when the real parts of the complex energy
levels are considered. We derive full many-body energy spectra of the fermionic
and bosonic Hatano-Nelson models as examples by filling single-particle energy
levels in the momentum space. For open boundary conditions, we show that
many-body non-Hermitian skin effects persist in all many-body eigenstates for
both fermions and bosons. Furthermore, we find surprisingly that the ground
state of bosons is an anomalous Bose-Einstein condensation with all of the
particles simultaneously localizing in both the real and momentum space beyond
the Heisenberg uncertainty principle. For periodic boundary conditions, we show
that hard-core bosons cannot be mapped to fermions. This work establishes a
general framework for understanding the many-body physics of non-Hermitian
systems, revealing rich unique non-Hermitian many-body physics.
