  We find the complete phase diagram of a generalised XY model that includes
half-vortices. The model possesses superfluid, pair-superfluid and disordered
phases, separated by Kosterlitz-Thouless (KT) transitions for both the
half-vortices and ordinary vortices, as well as an Ising-type transition. There
also occurs an unusual deconfining phase transition, where the disordered to
superfluid transition is of Ising rather than KT type. We show by analytical
arguments and extensive numerical simulations that there is a point in the
phase diagram where the KT transition line meets the deconfining Ising phase
transition. We find that the latter extends into the disordered phase not as a
phase transition, but rather solely as a deconfinement transition. It is best
understood in the dual height model, where on one side of the transition height
steps are bound into pairs while on the other they are unbound. We also extend
the phase diagram of the dual model, finding both O(2) loop model and
antiferromagnetic Ising transitions.
