  We construct nontrivial smooth bounded domains $\Omega \subseteq
\mathbb{R}^n$ of the form $\Omega_0 \setminus \overline{\Omega}_1$, bifurcating
from annuli, for which there exists a positive solution to the overdetermined
boundary value problem \[
  -\Delta u = 1, \; u>0 \quad \text{in } \Omega, \qquad
  u = 0 ,\; \partial_\nu u = \text{const} \quad \text{on } \partial\Omega_0,
\qquad
  u = \text{const} ,\; \partial_\nu u = \text{const} \quad \text{on } \partial
\Omega_1, \] where $\nu$ stands for the inner unit normal to $\partial\Omega$.
From results by Reichel and later by Sirakov, it was known that the condition
$\partial_\nu u \leq 0$ on $\partial\Omega_1$ is sufficient for rigidity to
hold, namely, the only domains which admit such a solution are annuli and
solutions are radially symmetric. Our construction shows that the condition is
also necessary. In addition, the constructed domains are shown to be
self-Cheeger.
