  Monitored quantum circuits allow for unprecedented dynamical control of
many-body entanglement. Here we show that random, measurement-only circuits,
implementing the competition of bond and plaquette couplings of the Kitaev
honeycomb model, give rise to a structured volume-law entangled phase with
subleading $L \ln L$ liquid scaling behavior. This interacting Majorana liquid
takes up a highly-symmetric, spherical parameter space within the entanglement
phase diagram obtained when varying the relative coupling probabilities. The
sphere itself is a critical boundary with quantum Lifshitz scaling separating
the volume-law phase from proximate area-law phases, a color code or a toric
code. An exception is a set of tricritical, self-dual points exhibiting
effective (1+1)d conformal scaling at which the volume-law phase and both
area-law phases meet. From a quantum information perspective, our results
define error thresholds for the color code in the presence of projective error
and stochastic syndrome measurements. We show that an alternative realization
of our model circuit can be implemented using unitary gates plus ancillary
single-qubit measurements only.
