We analyze the thermodynamics of the atomic and (nematic) pair superfluids appearing in the attractive two-dimensional Bose-Hubbard model with a three-body hard-core constraint that has been derived as an effective model for cold atoms subject to strong three-body losses in optical lattices. We show that the thermal disintegration of the pair superfluidity is governed by the proliferation of fractional half-vortices leading to a Berezinskii-Kosterlitz-Thousless transition with unusual jump in the helicity modulus. In addition to the (conventional) Berezinskii-Kosterlitz-Thousless transition out of the atomic superfluid, we furthermore identify a direct thermal phase transition separating the pair and the atomic superfluid phases, and show that this transition is continuous with critical scaling exponents consistent with those of the two-dimensional Ising universality class. We exhibit a direct connection between the partial loss of quasi long-range order at the Ising transition between the two superfluids and the parity selection in the atomic winding number fluctuations that distinguish the atomic from the pair superfluid.
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