  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 (DSFF). 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 $\mathbb{Z_2}$ symmetry, a modification of the measurement
basis from the transverse to the longitudinal direction makes the phase
transition disappear altogether.
