  Extensive theoretical and experimental investigation has been conducted on
fermion pair condensation and exciton condensation as distinct classes of
Bose-Einstein-like condensation. In this work, the existence of a
fermion-exciton condensate---a single quantum state in which character of both
fermion pair and exciton condensates coexist---is established computationally
in the low-particle-number ($N$) limit and theoretically in the large-$N$
thermodynamic limit. The trade-off between the fermion pair and excitonic
character of the fermion-exciton condensate is shown to be elliptic in nature.
The possibility that the properties of fermion-exciton condensates could be a
hybrid of the properties of fermion pair condensates and exciton condensates is
discussed, and future experimental and computational exploration of this new
class of condensate, which may potentially be realizable in a bilayer of
superconductors, is anticipated.
