  We study rare three-body decays of the Omega baryon using SU(3) chiral
perturbation theory, the successful effective field theory of quantum
chromodynamics at low energies. At leading order, we calculate the branching
fractions of the decay $\Omega^- \to \Xi \pi \pi$ for all possible combinations
of pions. For one channel we find an order-of-magnitude discrepancy between
theory and experiment. This tension is known to exist in the non-relativistic
limit, and we confirm that it remains in the relativistic calculation. Fairly
independent of the values of the low-energy constants we establish lower limits
for the branching fractions of these three-body Omega decays, which reaffirm
the gap between theory and experiment. We point out that this discrepancy is
closely tied to the $\Delta I =1/2$ selection rule. In turn, this means that
the three-body decays constitute an interesting tool to scrutinize the
selection rule. Using next-to-leading order calculations we also provide
predictions for the decay $\Omega^- \to \Xi^0 \mu^- \bar{\nu}_\mu$. We show
that fully-differential distributions will provide access to low-energy
constants needed in the axial-vector transitions from a decuplet to octet
baryon. Since data for all of these rare three-body Omega decays are scarce
(fully differential data are nonexistent), we recommend that they be remeasured
at running and upcoming experiments, such as BESIII, LHCb, Belle-II, and PANDA.
