Mesoscale eddies are found throughout the global ocean. Generally, they are referred to as “coherent” structures because they are organized rotating fluid elements that propagate within the ocean and have a long lifetime. Since in situ observations of the ocean are very rare, eddies have been characterized primarily from satellite observations or by relatively idealized approaches of geophysical fluid dynamics. Satellite observations provide access to only a limited number of surface features and exclusively for structures with a fingerprint on surface properties. Observations of the vertical sections of ocean eddies are rare. Therefore, important eddy properties, such as eddy transports or the characterization of eddy “coherence”, have typically been approximated by simple assumptions or by applying various criteria based on their velocity field or thermohaline properties. In this study, which is based on high-resolution in-situ data collection from the EUREC4A-OA field experiment, we show that Ertel potential vorticity is very appropriate to accurately identify the eddy core and its boundaries. This study provides evidence that the eddy boundaries are relatively intense and intimately related to both the presence of a different water mass in the eddy core from the background and to the isopycnal steepening caused by the volume of the eddy. We also provide a theoretical framework to examine their orders of magnitude and define an upper bound for the proposed definition of the eddy boundary. The results suggest that the eddy boundary is not a well-defined material boundary but rather a frontal region subject to instabilities.