  We consider the RSOS S-matrices of the Phi(1,5) perturbed minimal models
which have recently been found in the companion paper [hep-th/9604098]. These
S-matrices have some interesting properties, in particular, unitarity may be
broken in a stronger sense than seen before, while one of the three classes of
Phi(1,5) perturbations (to be described) shares the same Thermodynamic Bethe
Ansatz as a related Phi(1,2) perturbation. We test these new S-matrices by the
standard Truncated Conformal Space method, and further observe that in some
cases the BA equations for two particle energy levels may be continued to
complex rapidity to describe (a) single particle excitations and (b) complex
eigenvalues of the Hamiltonian corresponding to non-unitary S-matrix elements.
We make some comments on identities between characters in the two related
models following from the fact that the two perturbed theories share the same
breather sector.
