Why does every anyon need to have an antiparticle? It seems to be a basic requisite to every anyonic model that every type of anyon, say $a$ in the theory comes with an antiparticle $\bar{a}$ (which can be itself) where $a$ and $\bar{a}$ fuse together to the vacuum, i.e. have trivial statistics when viewed as a composite.
Why is the existence of an antiparticle a necessity for a sound anyonic theory?
 A: A anyon without anti-particle would be an allowed excitation on the plane, but not on the sphere (because all the excitations on a sphere have to fuse to the vacuum). So as long as you demand that the possible local excitations don't depend on the global topology, it must be the case that particles have anti-particles.
An example of a system where the allowed local excitations don't depend on the global topology is a Hamiltonian built of commuting local projectors. So at least any topological phase that can be realized by commuting projectors must have anti-particles.
More generally, (but less rigorously) you can try to argue from the point of view that a topological quantum field theory should be invariant under Euclidean rotations in space-time. So if you can have a world-line of particle $a$ pointing in the $+t$ direction, then you can also have a world-line pointing of $a$ pointing in the $-t$ direction, which is equivalent to a world-line of $\overline{a}$ in the $+t$ direction.
I think it's an interesting question, though, whether in non-relativistic systems you could get topological phases which are topological in terms of the spatial degrees of freedom, but don't necessarily allow for space and time to get rotated into each other. I believe some people would call these "H-type" theories (instead of "L-type") because they would be defined in terms of a topological Hamiltonian rather than a topological action/Lagrangian.
