  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.
