  We consider a two-dimensional electron gas (2DEG) in the Quantum Hall regime
in the presence of a Zeeman field, with the Fermi level tuned to filling factor
$\nu=1$. We show that, in the presence of spin-orbit coupling, contacting the
2DEG to a narrow strip of an s-wave superconductor produces a topological
superconducting gap along the contact as a result of crossed Andreev reflection
(CAR) processes across the strip. The sign of the topological gap, controlled
by the CAR amplitude, depends periodically on the Fermi wavelength and strip
width and can be externally tuned. An interface between two halves of a long
strip with topological gaps of opposite sign implements a robust
$\pi$-junction, hosting a pair of Majorana zero modes that do not split despite
their overlap. We show that such a situation can be exploited to perform
protected non-Abelian tunnel-braid operations without any fine tuning.
