  Motivated by the physics of coherently coupled, ultracold atom-molecule
mixtures, we investigate a classical model possessing the same symmetry --
namely a $U(1)\times \mathbb{Z}_2$ symmetry, associated with the mass
conservation in the mixture ($U(1)$ symmetry), times the $\mathbb{Z}_2$
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.
