We numerically investigate the robustness against various perturbations of measurement-induced phase transition in monitored quantum Ising models in the no-click limit, where the dynamics is described by a non-Hermitian Hamiltonian. We study perturbations that break the integrability and/or the symmetry of the model, as well as modifications in the measurement protocol, characterizing the resulting chaos and lack of integrability through the dissipative spectral form factor. We show that while the measurement-induced phase transition and its properties appear to be broadly insensitive to lack of integrability and breaking of the Z2 symmetry, a modification of the measurement basis from the transverse to the longitudinal direction makes the phase transition disappear altogether.
