Two-dimensional moiré materials provide a highly tunable platform to investigate strongly correlated electronic states. Such emergent many-body phenomena can be optically probed in moiré systems created by stacking two layers of transition metal dichalcogenide semiconductors: optically injected excitons can interact with itinerant carriers occupying narrow moiré bands to form exciton-polarons sensitive to strong correlations. Here, we investigate the behaviour of excitons dressed by a Fermi sea localised by the moiré superlattice of a molybdenum diselenide (MoSe₂)/tungsten diselenide (WSe₂) twisted hetero-bilayer. At a multitude of fractional fillings of the moiré lattice, we observe ordering of both electrons and holes into stable correlated electronic states. Magneto-optical measurements reveal extraordinary Zeeman splittings of the exciton-polarons due to exchange interactions in the correlated hole phases, with a maximum close to the correlated state at one hole per site. The temperature dependence of the Zeeman splitting reveals antiferromagnetic ordering of the correlated holes across a wide range of fractional fillings. Our results illustrate the nature of exciton-polarons in the presence of strongly correlated electronic states and reveal the rich potential of the MoSe₂/WSe₂ platform for investigations of Fermi–Hubbard and Bose–Hubbard physics.