  We consider conformal nets on $S^1$ of von Neumann algebras, acting on the
full Fock space, arising in free probability. These models are twisted local,
but non-local. We extend to the non-local case the general analysis of the
modular structure. The local algebras turn out to be $III_1$-factors associated
with free groups. We use our set up to show examples exhibiting arbitrarily
large maximal temperatures, but failing to satisfy the split property, then
clarifying the relation between the latter property and the trace class
conditions on $e^{-\b L}$, where $L$ is the conformal Hamiltonian.
