Motivated by the physics of coherently coupled, ultracold atom-molecule mixtures, we investigate a classical model possessing the same symmetry – namely a U(1)×Z2 symmetry, associated with the mass conservation in the mixture (U(1) symmetry), times the Z2 symmetry in the phase relationship between atoms and molecules.  

In two spatial dimensions the latter symmetry can lead to a finite-temperature  Ising  transition,  associated  with  (quasi)  phase  locking  between  the  atoms  and  the molecules.  On the other hand, the U(1) symmetry has an associated Berezinskii-Kosterlitz-Thouless (BKT)  transition  towards  quasi-condensation  of  atoms  or  molecules.   The  existence  of  the  two transitions is found to depend crucially on the population imbalance (or detuning) between atoms and  molecules:   when  the  molecules  are  majority  in  the  system,  their  BKT  quasi-condensation transition occurs at a higher temperature than that of the atoms; the latter has the unconventional nature of an Ising (quasi) phase-locking transition, lacking a finite local order parameter below the critical temperature.  When the balance is gradually biased towards the atoms, the two transitions merge together to leave out a unique BKT transition, at which both atoms and molecules acquire quasi-long-range correlations, but only atoms exhibit conventional BKT criticality, with binding of vortex-antivortex pairs into short-range dipoles.  The molecular vortex-antivortex excitations bind as well, but undergo a marked crossover from a high-temperature regime in which they are weakly bound, to a low-temperature regime of strong binding, reminiscent of their transition in the absence of atom-molecule coupling.