  We study the quantum dynamics of a many-body system subject to coherent
evolution and coupled to a non-Markovian bath. We propose a technique to
unravel the non-Markovian dynamics in terms of quantum jumps, a connection that
was so far only understood for single-body systems. We develop a systematic
method to calculate the probability of a quantum trajectory, and formulate it
in a diagrammatic structure. We find that non-Markovianity renormalizes the
probability of realizing a quantum trajectory, and that memory effects can be
interpreted as a perturbation on top of the Markovian dynamics. We show that
the diagrammatic structure is akin to that of a Dyson equation, and that the
probability of the trajectories can be calculated analytically. We then apply
our results to study the measurement-induced entanglement transition in random
unitary circuits. We find that non-Markovianity does not significantly shift
the transition, but stabilizes the volume law phase of the entanglement by
shielding it from transient strong dissipation.
