  A strong low-temperature enhancement of the tunneling conductance between
graphene bilayers has been reported recently, and interpreted as a signature of
equilibrium electron-hole pairing, first predicted in bilayers more than forty
years ago but previously unobserved. Here we provide a detailed theory of
conductance enhanced by fluctuating electron-hole Cooper pairs, which are a
precursor to equilibrium pairing, that accounts for specific details of the
multi-band double graphene bilayer system which supports several different
pairing channels. Above the equilibrium condensation temperature, pairs have
finite temporal coherence and do not support dissipationless tunneling.
Instead, they strongly boost the tunneling conductivity via a fluctuational
internal Josephson effect. Our theory makes predictions for the dependence of
the zero bias peak in the differential tunneling conductance on temperature and
electron-hole density imbalance that capture important aspects of the
experimental observations. In our interpretation of the observations, cleaner
samples with longer disorder scattering times would condense at temperatures
$T_c$ up to $\sim 50 {\rm K}$, compared to the record $T_c \sim 1.5 $K achieved
to date in the experiment.
