  We study a $U(1)\times U(1)$ system with short-range interactions and mutual
$\theta=2\pi/3$ statistics in (2+1) dimensions. We are able to reformulate the
model to eliminate the sign problem, and perform a Monte Carlo study. We find a
phase diagram containing a phase with only small loops and two phases with one
species of proliferated loop. We also find a phase where both species of loop
condense, but without any gapless modes. Lastly, when the energy cost of loops
becomes small we find a phase which is a condensate of bound states, each made
up of three particles of one species and a vortex of the other. We define
several exact reformulations of the model, which allow us to precisely describe
each phase in terms of gapped excitations. We propose field-theoretic
descriptions of the phases and phase transitions, which are particularly
interesting on the "self-dual" line where both species have identical
interactions. We also define irreducible responses useful for describing the
phases.
