We assume that
Ω
1
,
Ω
2
⊂
R
n
+
1
\Omega _1, \Omega _2 \subset \mathbb {R}^{n+1}
,
n
≥
1
n \geq 1
, are two disjoint domains whose complements satisfy the capacity density condition and where the intersection of their boundaries
F
F
has positive harmonic measure. Then we show that in a fixed ball
B
B
centered on
F
F
, if the harmonic measure of
Ω
1
\Omega _1
satisfies a scale invariant
A
∞
A_\infty
-type condition with respect to the harmonic measure of
Ω
2
\Omega _2
in
B
B
, then there exists a uniformly
n
n
-rectifiable set
Σ
\Sigma
so that the harmonic measure of
Σ
∩
F
\Sigma \cap F
contained in
B
B
is bounded below by a fixed constant independent of
B
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
Ω
1
\Omega _1
and
Ω
2
\Omega _2
are complementary NTA domains, we obtain a characterization of the
A
∞
A_\infty
condition between the respective harmonic measures of
Ω
1
\Omega _1
and
Ω
2
\Omega _2
.