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.Driven dissipative systems are ubiquitous in many body physics and cavity QED [1][2][3][4][5][6][7][8][9][10][11][12]. These systems are typically gapped and feature a unique, non-equilibrium steady state. In the regime of a dissipative phase transition (DPT), however, this gap vanishes and the null-space of the Liouvillian is spanned by several compatible steady-states. [13][14][15][16][17][18][19][20][21][22]. Due to their fundamental interest and practical applications, such as enhanced metrological properties [23,24], DPTs have attracted a significant amount of attention, with much work being devoted to study the associated phenomena of bistability [3-5, 22, 25-28], hysteresis [2,29], intermittency [6,26,[29][30][31][32], multimodality [25,31], metastability [33] and symmetry breaking [34][35][36]. All these effects are understood as different manifestations of the coexistence of several non-equilibrium phases. In particular, many experiments will look for intermittency as the hallmark of such phase coexistence [6,26,[29][30][31][32]. Intermittency is a phenomenon defined by a random switching between periods of high and low dynamical activity (for instance in the rate of photon emission). This behaviour, which is observed during a single run of the experiment, is conveniently described using the formalism of quantum jumps in which the system is characterized in terms of a pure wavefunction that undergoes stochastic evolution [37][38][39].The timescale Ï of this intermittency is given by the inverse of the Liouvillian gap or asymptotic decay rate (ADR), i.e. the eigenvalue λ 2 of the Liouvillian operator L with the second largest real part [23,40,41]. Since a DPT is defined by a vanishing Liouvillian gap [13,14], it will necessarily imply that Ï diverges. In most typical situations, this closing is reached in the thermod...