  Repeated measurements can induce entanglement phase transitions in the
dynamics of quantum systems. Interacting models, both chaotic and integrable,
generically show a stable volume-law entangled phase at low measurement rates
which disappears for free, Gaussian fermions. Interactions break the
Gaussianity of a dynamical map in its unitary part, but non-Gaussianity can be
introduced through measurements as well. By comparing the entanglement and
non-Gaussianity structure of different protocols, we propose a new
single-particle indicator of the measurement-induced phase transition and we
use it to argue in favour of the stability of the transition when
non-Gaussianity is purely provided by measurements
