  The interplay of unitary evolution and local measurements in many-body
systems gives rise to a stochastic state evolution and to measurement-induced
phase transitions in the pure state entanglement. In realistic settings,
however, this dynamics may be spoiled by decoherence, e.g., dephasing, due to
coupling to an environment or measurement imperfections. We investigate the
impact of dephasing and the inevitable evolution into a non-Gaussian, mixed
state, on the dynamics of monitored fermions. We approach it from three
complementary perspectives: (i) the exact solution of the conditional master
equation for small systems, (ii) quantum trajectory simulations of Gaussian
states for large systems, and (iii) a renormalization group analysis of a
bosonic replica field theory. For weak dephasing, constant monitoring preserves
a weakly mixed state, which displays a robust measurement-induced phase
transition between a critical and a pinned phase, as in the decoherence-free
case. At strong dephasing, we observe the emergence of a new scale describing
an effective temperature, which is accompanied with an increased mixedness of
the fermion density matrix. Remarkably, observables such as density-density
correlation functions or the subsystem parity still display scale invariant
behavior even in this strongly mixed phase. We interpret this as a signature of
gapless, classical diffusion, which is stabilized by the balanced interplay of
Hamiltonian dynamics, measurements, and decoherence.
