  In quantum theory, particles in three spatial dimensions come in two
different types: bosons or fermions, which exhibit sharply contrasting
behaviours due to their different exchange statistics. Could more general forms
of probabilistic theories admit more exotic types of particles? Here, we
propose a thought experiment to identify more exotic particles in general
post-quantum theories. We consider how in quantum theory the phase introduced
by swapping indistinguishable particles can be measured. We generalise this to
post-quantum scenarios whilst imposing indistinguishability and locality
principles. We show that our ability to witness exotic particle exchange
statistics depends on which symmetries are admitted within a theory. These
exotic particles can manifest unusual behaviour, such as non-abelianicity even
in topologically simple three-dimensional space.
