In dissipative quantum systems, strong symmetries can lead to the existence of conservation laws and multiple steady states. The investigation of such strong symmetries and their consequences on the dynamics of the dissipative systems is still in its infancy. In this work we investigate a strong symmetry for bosonic atoms coupled to an optical cavity, an experimentally relevant system, using adiabatic elimination techniques and numerically exact matrix product state methods. We show the existence of multiple steady states for ideal bosons coupled to the cavity. We find that the introduction of a weak breaking of the strong symmetry by a small interaction term leads to a direct transition from multiple steady states to a unique steady state. We point out the phenomenon of dissipative freezing, the breaking of the conservation law at the level of individual realizations in the presence of the strong symmetry. For a weak breaking of the strong symmetry we see that the behavior of the individual trajectories still shows some signs of this dissipative freezing before it fades out for a larger symmetry breaking terms.
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