Two glider transects in the Gulf of Mexico reveal fine-vertical-scale thermohaline structures within a Loop Current eddy (LCE). Partially compensating temperature and salinity anomalies are shown to organize as thin layers below the eddy and near its edges. The anomalies have vertical scales ranging from 2 to 60 m and extend laterally over distances up to 120 km. These structures are evident in synthetic acoustic reflectivity derived from the glider data and are reminiscent of the intense layering observed in seismic imagery around meddies, Agulhas rings, and warm-core Kuroshio rings. The observed layers are aligned with the geostrophic streamfunction rather than isopycnals and develop preferentially in zones of intense vertical shear. These observations suggest that tracer stirring by the eddy's vertically sheared azimuthal flow might be an important process for their generation. In an attempt to rationalize this process, high-resolution quasigeostrophic simulations were performed using an idealized anticyclonic ring for the initial conditions. As the vortex destabilizes, layering rapidly develops in the model, resulting in structures similar to those found in the observation data. Passive tracer experiments also suggest that the layers form through differential advection of the tracer field by the vertically sheared flow associated with the LCE.