  We propose a $\mathbb{U}(1) \times \mathbb{Z}_2$ effective gauge theory for
vortices in a $p_x+ip_y$ superfluid in two dimensions. The combined gauge
transformation binds $\mathbb{U}(1)$ and $\mathbb{Z}_2$ defects so that the
total transformation remains single-valued and manifestly preserves the the
particle-hole symmetry of the action. The $\mathbb{Z}_2$ gauge field introduces
a complete Chern-Simons term in addition to a partial one associated with the
$\mathbb{U}(1)$ gauge field. The theory reproduces the known physics of vortex
dynamics such as a Magnus force proportional to the superfluid density. More
importantly, it predicts a universal Abelian phase, $\exp(i\pi/8)$, upon the
exchange of two vortices. This phase is modified by non-universal corrections
due to the partial Chern-Simon term, which are nevertheless screened in a
charged superfluid at distances that are larger than the penetration depth.
