Strongly correlated bosons in a lattice are a platform that can realize rich bosonic states of matter and quantum phase transitions1. While strongly correlated bosons in a lattice have been studied in cold-atom experiments2–4, their realization in a solid-state system has remained challenging5. Here we trap interlayer excitons–bosons composed of bound electron–hole pairs, in a lattice provided by an angle-aligned WS2/bilayer WSe2/WS2 multilayer. The heterostructure supports Coulomb-coupled triangular moiré lattices of nearly identical period at the top and bottom interfaces. We observe correlated insulating states when the combined electron filling factor of the two lattices, with arbitrary partitions, equals 13,23,43 and 53. These states can be interpreted as exciton density waves in a Bose–Fermi mixture of excitons and holes6,7. Because of the strong repulsive interactions between the constituents, the holes form robust generalized Wigner crystals8–11, which restrict the exciton fluid to channels that spontaneously break the translational symmetry of the lattice. Our results demonstrate that Coulomb-coupled moiré lattices are fertile ground for correlated many-boson phenomena12,13. © 2023, The Author(s), under exclusive licence to Springer Nature Limited.