  We study the transient dynamics of an $A+B \rightarrow 0$ process on a pair
of randomly coupled networks, where reactants are initially separated. We find
that, for sufficiently small fractions $q$ of cross-couplings, the
concentration of $A$ (or $B$) particles decays linearly in a first stage and
crosses over to a second linear decrease at a mixing time $t_x$. By numerical
and analytical arguments, we show that for symmetric and homogeneous structures
$t_x\propto(\nicefrac{\langle k \rangle}{q})\log(\nicefrac{\langle k
\rangle}{q})$ where $\langle k \rangle$ is the mean degree of both networks.
Being this behavior in marked contrast with a purely diffusive process---where
the mixing time would go simply like $\langle k\rangle/q$---we identify the
logarithmic slowing down in $t_x$ to be the result of a novel spontaneous
mechanism of {\em repulsion} between the reactants $A$ and $B$ due to the
interactions taking place at the networks' interface. We show numerically how
this spontaneous repulsion effect depends on the topology of the underlying
networks.
