  We study the incoherent transport of bosonic particles through a one
dimensional lattice with different left and right hopping rates, as modelled by
the asymmetric simple inclusion process (ASIP). Specifically, we show that as
the current passing through this system increases, a transition occurs, which
is signified by the appearance of a characteristic zigzag pattern in the
stationary density profile near the boundary. In this highly unusual transport
phase, the local particle distribution alternates on every site between a
thermal distribution and a Bose-condensed state with broken U(1)-symmetry.
Furthermore, we show that the onset of this phase is closely related to the
so-called non-Hermitian skin effect and coincides with an exceptional point in
the spectrum of density fluctuations. Therefore, this effect establishes a
direct connection between quantum transport, non-equilibrium condensation
phenomena and non-Hermitian topology, which can be probed in cold-atom
experiments or in systems with long-lived photonic, polaritonic and plasmonic
excitations.
