Recently there has been an intense effort to understand measurement induced
transitions, but we still lack a good understanding of non-Markovian effects on
these phenomena. To that end, we consider two coupled chains of free fermions,
one acting as the system of interest, and one as a bath. The bath chain is
subject to Markovian measurements, resulting in an effective non-Markovian
dissipative dynamics acting on the system chain which is still amenable to
numerical studies in terms of quantum trajectories. Within this setting, we
study the entanglement within the system chain, and use it to characterize the
phase diagram depending on the ladder hopping parameters and on the measurement
probability. For the case of pure state evolution, the system is in an area law
phase when the internal hopping of the bath chain is small, while a non-area
law phase appears when the dynamics of the bath is fast. The non-area law
exhibits a logarithmic scaling of the entropy compatible with a conformal
phase, but also displays linear corrections for the finite system sizes we can
study. For the case of mixed state evolution, we instead observe regions with
both area, and non-area scaling of the entanglement negativity. We quantify the
non-Markovianity of the system chain dynamics and find that for the regimes of
parameters we study, a stronger non-Markovianity is associated to a larger
entanglement within the system