The phase structure and renormalization group behavior of the XY model with p -fold random anisotropy is studied via time-dependent Langevin formulation. The statics and dynamics of the model are derived in two dimensions and extended to 2 + ϵ dimensions without the use of replicas. The connection with previous replica treatments is discussed. In two dimensions and above we show that a would-be spin glass phase is destroyed at large distances by the topological defects viz. vortices. The mechanism is: random interactions generate random Dzyaloshinskii-Moriya bond interactions which unbind the vortices and this in turn renders the glass phase paramagmetic at large scales. In two dimensions and p 2 > 8 there is a re-entrant transition from an intermediate XY phase into the glass phase. In 2 + ϵ dimensions and p 2 > 8 there is a transition governed by the pure XY fixed point from a paramagnetic phase into a low-temperature glass phase, paramagnetic at large length scales.