  In driven-dissipative systems, the presence of a strong symmetry guarantees
the existence of several steady states belonging to different symmetry sectors.
Here we show that, when a system with a strong symmetry is initialized in a
quantum superposition involving several of these sectors, each individual
stochastic trajectory will randomly select a single one of them and remain
there for the rest of the evolution. Since a strong symmetry implies a
conservation law for the corresponding symmetry operator on the ensemble level,
this selection of a single sector from an initial superposition entails a
breakdown of this conservation law at the level of individual realizations.
Given that such a superposition is impossible in a classical, stochastic
trajectory, this is a a purely quantum effect with no classical analogue. Our
results show that a system with a closed Liouvillian gap may exhibit, when
monitored over a single run of an experiment, a behaviour completely opposite
to the usual notion of dynamical phase coexistence and intermittency, which are
typically considered hallmarks of a dissipative phase transition. We discuss
our results with a simple, realistic model of squeezed superradiance.
