  We study a 2D measurement-only random circuit motivated by the Bacon-Shor
error correcting code. We find a rich phase diagram as one varies the relative
probabilities of measuring nearest neighbor Pauli XX and ZZ check operators. In
the Bacon-Shor code, these checks commute with a group of stabilizer and
logical operators, which therefore represent conserved quantities. Described as
a subsystem symmetry, these conservation laws lead to a continuous phase
transition between an X-basis and Z-basis spin glass order. The two phases are
separated by a critical point where the entanglement entropy between two halves
of an L X L system scales as L ln L, a logarithmic violation of the area law.
We generalize to a model where the check operators break the subsystem
symmetries (and the Bacon-Shor code structure). In tension with established
heuristics, we find that the phase transition is replaced by a smooth
crossover, and the X- and Z-basis spin glass orders spatially coexist.
Additionally, if we approach the line of subsystem symmetries away from the
critical point in the phase diagram, some spin glass order parameters jump
discontinuously
