  Non-stationary long-time dynamics was recently observed in a driven
two-component Bose-Einstein condensate coupled to an optical cavity [N. Dogra,
et al. arXiv:1901.05974] and analyzed in mean-field theory. We solve the
underlying model in the thermodynamic limit and show that this system is always
dynamically unstable -- even when mean-field theory predicts stability.
Instabilities always occur in higher-order correlation functions leading to
squeezing and entanglement induced by cavity dissipation. The dynamics may be
understood as the formation of a dissipative time crystal. We use perturbation
theory for finite system sizes to confirm the non-stationary behaviour.
