Coupled two-dimensional electron-hole bilayers provide a unique platform to study strongly correlated Bose-Fermi mixtures in condensed matter. Electrons and holes in spatially separated layers can bind to form interlayer excitons, composite Bosons expected to support high-temperature exciton superfluids. The interlayer excitons can also interact strongly with excess charge carriers when electron and hole densities are unequal. Here, we use optical spectroscopy to quantitatively probe the local thermodynamic properties of strongly correlated electron-hole fluids in MoSe2/hBN/WSe2 heterostructures. We observe a discontinuity in the electron and hole chemical potentials at matched electron and hole densities, a definitive signature of an excitonic insulator ground state. The excitonic insulator is stable up to a Mott density of ~$0.8\times {10}^{12} \mathrm{cm}^{-2}$ and has a thermal ionization temperature of ~70 K. The density dependence of the electron, hole, and exciton chemical potentials reveals strong correlation effects across the phase diagram. Compared with a non-interacting uniform charge distribution, the correlation effects lead to significant attractive exciton-exciton and exciton-charge interactions in the electron-hole fluid. Our work highlights the unique quantum behavior that can emerge in strongly correlated electron-hole systems.
