For example, both the $\nu=1/3$ Laughlin state and the Moore-Read state has a simple interpretation in terms of composite fermions, which are bound states of an electron and two fluxes.
Both the Laughlin states and the Moore-Read state also have anyons, since they are both topologically ordered. Laughlin states have Abelian $ne/m$ anyons, with $m=1/\nu$ and $n<m$, and Moore-Read state hosts non-Abelian anyons $\sigma$ with charge $e/4$ and a neutral fermion $\chi$.
However, composite fermions themselves do not appear in the anyon contents of either state, despite being such an important step in describing these states. My question is why.