  We study bicolored configurations of points in the Euclidean $n$-space that
are constrained to remain either inside or outside a fixed Euclidean
$m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of
the Swiss-Cheese operad, called the complementarily constrained disks operad
$\mathsf{CD}_{mn}$, associated to such configurations. The operad
$\mathsf{CD}_{mn}$ is weakly equivalent to the operad of locally constant
factorization algebras on the stratified space $\{\mathbb{R}^{m} \subset
\mathbb{R}^{n}\}$. We prove that this operad is formal over $\mathbb{R}$.
