A Monte Carlo algorithm has been used to study the XY model with random threefold single-site anisotropy on simple cubic lattices, in the strong-anisotropy limit. The model has a pure XY-like ferromagnetic critical point at ${\mathit{T}}_{\mathit{c}}$/J=2.190±0.005. The zero-field thermodynamic functions remain close to those of the simple XY model down to T≊2.00J. The ground state has an energy per spin of ${\mathit{E}}_0$/J=-2.2305±0.0010 and a magnetization per spin of ${\mathit{M}}_0$=0.720±0.003.

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