  I study the effects of particle-hole imbalance on the exciton superfluid
formed in a topological insulator thin-film and obtain the mean-field phase
diagram. At finite imbalance a spatially modulated condensate is formed, akin
to the Fulde-Ferrel-Larkin-Ovchinikov state in a superconductor, which preempts
a first-order transition from the uniform condensate to the normal state at low
temperatures. The imbalance can be tuned by changing the chemical potential at
the two surfaces separately or, alternatively, by an asymmetric application of
Zeeman fields at constant chemical potential. A vortex in the condensate
carries a precisely fractional charge half of that of an electron. Possible
experimental signatures for realistic parameters are discussed.
