Mesoscale eddies are a ubiquitous feature of the global ocean. According to Lagrangian theory, these eddies often transport a distinct water mass within their cores, making them materially coherent. This study aims to determine if such a distinct water mass exists in eddy cores, thereby verifying their material coherence using in situ data, despite the lack of temporal continuity. We introduce the term "thermohaline coherence" to describe this approach. Identifying such a water mass would signal Lagrangian transport from the eddy formation region. We analyzed the water masses at the cores of various eddies sampled during eight research cruises, using high-resolution data (approximately 20 km horizontally and 10 m vertically). We revisited coherence definitions and checked data accuracy. Comparing the horizontal positions of these core anomalies with eddy surface signatures revealed that surface data alone are insufficient for characterizing the eddy material coherence. To calculate eddy volumes, we compare thermohaline anomalies with other criteria and we present two methods for extrapolating eddy volumes from a single hydrographic section. The results show that the outermost closed contour of the Brunt-Väisälä frequency anomaly at each depth provides a reliable approximation for the eddy boundary.