  Engineering experimentally accessible solid state platforms that facilitate
the realization of Majorana zero modes (MZMs) has been a challenging task over
recent years. Here we employ a complex edge consisting of both "downstream" and
"upstream" chiral modes to design a MZM-supporting platform, and show that the
latter does not necessitate the introduction of an external superconducting
contact. Specifically we consider the edge of a $\nu=2/3$ fractional quantum
Hall phase, where the downstream (upstream) mode is charged (neutral). We show
that the boundary of an elongated quantum anti-dot ("quantum anti-wire")
embedded in the $\nu =2/3$ phase can host MZMs in the neutral sector. The
emergence of the MZMs is induced by gap-opening processes along the quantum
anti-wire, which are incompatible with gap-opening processes in the bulk of the
$\nu=2/3$ state. An experimental setup to identify neutral MZMs is proposed.
