We report the discovery of an H-alpha-emitting bow-shock nebula powered by the nearby millisecond pulsar J2124-3358. The bow shock is very broad, and is highly asymmetric about the pulsars velocity vector. This shape is not consistent with that expected for the case of an isotropic wind interacting with a homogeneous ambient medium. Models which invoke an anisotropy in the pulsar wind, a bulk flow of the surrounding gas, or a density gradient in the ambient medium either perpendicular or parallel to the pulsars direction of motion also fail to reproduce the observed morphology. However, we find an ensemble of good fits to the nebular morphology when we consider a combination of these effects. In all such cases, we find that the pulsar is propagating through an ambient medium of mean density 0.8-1.3 cm^(-3) and bulk flow velocity ~15-25 km/s and that the star has recently encountered an increase in density by 1-10 cm^(-3) over a scale ~<0.02 pc. The wide variety of models which fit the data demonstrate that in general there is no unique set of parameters which can be inferred from the morphology of a bow-shock nebula.
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