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We consider an electron-hole bilayer in the limit of extreme densityimbalance, where we have a single particle in one layer interactingattractively with a Fermi liquid in the other parallel layer. Using anappropriate variational wave function for the dressed exciton, we providestrong evidence for the existence of the Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) phase in electron-hole bilayers with a large density imbalance.Furthermore, within this unusual limit of FFLO, we find that a dilute gas ofminority particles forms excitons that condense into a two-dimensional"supersolid".

Supersolidity in electron-hole bilayers with a large density imbalance