  The state with a spontaneous interlayer phase coherence in a graphene based
bilayer quantum Hall system is studied. This state can be considered as a gas
of superfluid electron-hole pairs with the components of the pair belonging to
different layers. Superfluid flux of such pairs is equivalent to two electrical
supercurrents in the layers. It is shown that the state with the interlayer
phase coherence emerges in the graphene system if a certain imbalance of the
Landau level filling factors of the layers is created. We obtain the
temperature of transition into the superfluid state, the maximum interlayer
distance at which the phase coherence is possible, and the critical values of
the supercurrent. The advantages of use of graphene systems instead of GaAs
heterostructures for the realization of the bilayer electron-hole
superconductivity is discussed.
