  We introduce and investigate an open many-body quantum system in which
kinetically constrained coherent and dissipative processes compete. The form of
the incoherent dissipative dynamics is inspired by that of epidemic spreading
or cellular-automaton-based computation related to the density-classification
problem. It features two non-fluctuating absorbing states as well as a
$\mathcal{Z}_2$-symmetric point in parameter space. The coherent evolution is
governed by a kinetically constrained $\mathcal{Z}_2$-symmetric many-body
Hamiltonian which is related to the quantum XOR-Fredrickson-Andersen model. We
show that the quantum coherent dynamics can stabilize a fluctuating state and
we characterize the transition between this active phase and the absorbing
states. We also identify a rather peculiar behavior at the
$\mathcal{Z}_2$-symmetric point. Here the system approaches the absorbing-state
manifold with a dynamics that follows a power-law whose exponent continuously
varies with the relative strength of the coherent dynamics. Our work shows how
the interplay between coherent and dissipative processes as well as symmetry
constraints may lead to a highly intricate non-equilibrium evolution and may
stabilize phases that are absent in related classical problems.
