  In this Letter, we report the onset of periodic oscillations of coherences in
an interacting bosonic gas coupled to a resonator after a quantum quench. This
dynamics extends the collapse and revival features of atomic correlations in
optical lattices to a dissipative scenario and exhibits hallmarks of
synchronization. The behavior emerges from the interplay of the quantum
dissipative nature of the cavity field and the presence of a (approximate)
strong symmetry in the dissipative system, providing a general recipe to
engineer intriguing quantum dynamics. Additionally, we show that the
approximate symmetry can arise dynamically during self-organization and can be
employed to obtain long-lived coherences.
