  We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are
two disjoint domains whose complements satisfy the capacity density condition
and the intersection of their boundaries $F$ has positive harmonic measure.
Then we show that in a fixed ball $B$ centered on $F$, if the harmonic measure
of $\Omega_1$ satisfies a scale invariant $A_\infty$-type condition with
respect to the harmonic measure of $\Omega_2$ in $B$, then there exists a
uniformly $n$-rectifiable set $\Sigma$ so that the harmonic measure of $\Sigma
\cap F$ contained in $B$ is bounded below by a fixed constant independent of
$B$. A remarkable feature of this result is that the harmonic measures do not
need to satisfy any doubling condition. In the particular case that $\Omega_1$
and $\Omega_2$ are complementary NTA domains, we obtain a geometric
characterization of the $A_\infty$ condition between the respective harmonic
harmonic measures of $\Omega_1$ and $\Omega_2$.
