  An exotic feature of the fractional quantum Hall effect is the emergence of
anyons, which are quasiparticle excitations with fractional statistics. In the
presence of a symmetry, such as $U(1)$ charge conservation, it is well known
that anyons can carry fractional symmetry quantum numbers. In this work we
reveal a different class of symmetry realizations: i.e. anyons can "breed" in
multiples under symmetry operation. We focus on the global Ising ($Z_2$)
symmetry and show examples of these unconventional symmetry realizations in
Laughlin-type fractional quantum Hall states. One remarkable consequence of
such an Ising symmetry is the emergence of anyons on the Ising symmetry domain
walls. We also provide a mathematical framework which generalizes this
phenomenon to any Abelian topological orders.
