  We study dynamical phase transitions occurring in the stationary state of the
dynamics of integrable many-body non-hermitian Hamiltonians, which can be
either realized as a no-click limit of a stochastic Schr\"{o}dinger equation or
using spacetime duality of quantum circuits. In two specific models, the
Transverse Field Ising Chain and the Long Range Kitaev Chain, we observe that
the entanglement phase transitions occurring in the stationary state have the
same nature as that occurring in the vacuum of the non-hermitian Hamiltonian:
bounded entanglement entropy when the imaginary part of the quasi-particle
spectrum is gapped and a logarithmic growth for gapless imaginary spectrum.
This observation suggests the possibility to generalize the area-law theorem to
non-Hermitian Hamiltonians.
