  We investigate the composite systems consisting of topological orders
separated by gapped domain walls. We derive a pair of domain-wall Verlinde
formulae, that elucidate the connection between the braiding of interdomain
excitations labeled by pairs of anyons in different domains and quasiparticles
in the gapped domain wall with their respective fusion rules. Through explicit
non-Abelian examples, we showcase the calculation of such braiding and fusion,
revealing that the fusion rules for interdomain excitations are generally
fractional or irrational. By investigating the correspondence between composite
systems and anyon condensation, we unveil the reason for designating these
fusion rules as symmetry fractionalized (irrationalized) fusion rules. Our
findings hold promise for applications across various fields, such as
topological quantum computation, topological field theory, and conformal field
theory.
