  We study measurement-induced symmetry-protected topological (SPT) order in a
wide class of quantum random circuit models by combining calculations within
the stabilizer formalism with tensor network simulations. We construct a family
of quantum random circuits, generating the out-of-equilibrium version of all
generalized cluster models, and derive a set of non-local string order
parameters to distinguish different SPT phases. We apply this framework to
investigate a random circuit realization of the XZX cluster model, and use the
string order parameter to demonstrate that the phase diagram is stable against
extending the class of unitary gates in the circuit, from Clifford gates to
Haar unitaries. We then turn to the XZZX generalized cluster model, and
demonstrate the coexistence of SPT order and spontaneous symmetry breaking, by
relying on string order parameters and a connected correlation function.
