The answer http://physics.stackexchange.com/a/43770 is interesting.
Suppose we have an anyon with a spin $p/q$ with relatively prime p,q , with an odd q. Suppose we have a bound state of $Nq$ such anyons where $N$ is an integer. This has to be bosonic. Suppose further this bound state is unstable and decays away completely into bosonic phonons and photons. The total angular momentum of the decay products has to remain the same due to conservation.
Suppose we have an anyon of the same species very far away orbiting around this bound state. The encircling phase factor means when we compute the total angular momentum of the system, it's the sum of the bosonic spin of the bound state, plus p/q of the faraway anyon, plus the relative orbital angular momentum plus $2N^2pq$, which is integral. That's before the decay.
This is a local theory. So, the very faraway anyon shouldn't be affected by the decay right away. This means its relative orbital angular momentum shouldn't change. However, there is no encircling phase factor for phonons and photons. Similarly, locality means the decay of the bound state shouldn't be affected by the presence/absence of a faraway anyon.
Don't tell me the total angular momentum of the system suddenly jumped by $2N^2pq$. On the other hand, don't tell me the relative orbital angular momentum jumped by $2N^2pq$ either.