In this talk we present the study of the long-time behaviour of a stochastic Allen-Cahn- Navier-Stokes system. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation and one on the phase variable in the Allen-Cahn equation, and includes the thermodynamically-relevant Flory-Huggins logarithmic potential. We first show existence of ergodic invariant measures. Secondly, we prove that if the noise acting in the Navier-Stokes equation is non-degenerate along a sufficiently large number of low modes, and the Allen-Cahn equation is highly dissipative, then the stochastic flow admits a unique invariant measure which is asymptotically stable with respect to a suitable Wasserstein metric. The talk is based on a joint work with A. Di Primio and L. Scarpa.