Comparison with Gaia MW GCs: normalized total energy as a function of the normalized z-component of the angular momentum at z = 0 for both in situ and ex situ GCs for the 198 MWs in the full MW potential + dynamical friction. The decision boundaries in dashed lines corresponding to probabilities of 0.05 (magenta), 0.5 (black), and 0.95 (green) are plotted to illustrate the separation between the two populations. Nephele GCs are the six GCs brought by the progenitor of Omega Centauri identified by Pagnini et al. (2025). The 0.5 boundary results in an error of 1.35% for in situ GCs and 11.2% for ex situ GCs. The extreme boundary at 0.95 (0.05) excludes 99.72% (97.3%) of in situ (ex situ) GCs while identifying 81.2% (76.4%) of ex situ (in situ) GCs. We establish that between these two limits, distinguishing the two populations is not feasible with previous methods due to the presence of a mixing zone. The orange dashed line represents the McMillan version of Belokurov & Kravtsov (2024) limit.