  We present and solve a model for the vortex configuration of a disordered
quantum Hall bilayer in the limit of strong and smooth disorder. We argue that
there is a characteristic disorder strength below which vortices will be rare,
and above which they proliferate. We predict that this can be observed tuning
the electron density in a given sample. The ground state in the strong-disorder
regime can be understood as an emulsion of vortex-antivortex crystals. Its
signatures include a suppression of the spatial decay of counterflow currents.
We find an increase of at least an order of magnitude in the length scale for
this decay compared to a clean system. This provides a possible explanation of
the apparent absence of leakage of counterflow currents through interlayer
tunneling, even in experiments performed deep in the coherent phase where
enhanced interlayer tunneling is observed.
