  Monitored quantum systems undergo Measurement-induced Phase Transitions
(MiPTs) stemming from the interplay between measurements and unitary dynamics.
When the detector readout is post-selected to match a given value, the dynamics
is generated by a Non-Hermitian Hamiltonian with MiPTs characterized by
different universal features. Here, we derive a partial post-selected
stochastic Schr\"odinger equation based on a microscopic description of
continuous weak measurement. This formalism connects the monitored and
post-selected dynamics to a broader family of stochastic evolution. We apply
the formalism to a chain of free fermions subject to partial post-selected
monitoring of local fermion parities. Within a 2-replica approach, we obtained
an effective bosonized Hamiltonian in the strong post-selected limit. Using a
renormalization group analysis, we find that the universality of the
non-Hermitian MiPT is stable against a finite (weak) amount of stochasticity.
We further show that the passage to the monitored universality occurs abruptly
at finite partial post-selection, which we confirm from the numerical finite
size scaling of the MiPT. Our approach establishes a way to study MiPTs for
arbitrary subsets of quantum trajectories and provides a potential route to
tackle the experimental post-selected problem.
