  Dissipation is inevitable in realistic quantum circuits. We examine the
effects of dissipation on a class of monitored random circuits that exhibit a
measurement-induced entanglement phase transition. This transition has
previously been understood as an order-to-disorder transition of an effective
classical spin model. We extend this mapping to include on-site dissipation
described by the dephasing and spontaneous emission channel and study the
corresponding 2D Ising model with $\mathbb{Z}_2$-symmetry-breaking
interactions. We analyze the dynamical regimes of the mutual information and
find that the joint action of monitored measurements and dissipation yields
short time, intermediate time and steady state behavior that can be understood
in terms of crossovers between different classical domain wall configurations.
The presented analysis applies to monitored open or Lindbladian quantum systems
and provides a tool to understand entanglement dynamics in realistic
dissipative settings and small achievable system sizes.
