  The constraints implied by analyticity in two-dimensional factorised S-matrix
theories are reviewed. Whenever the theory is not time-reversal invariant, it
is argued that the familiar condition of `Real analyticity' for the S-matrix
amplitudes has to be superseded by a different one known as `Hermitian
analyticity'. Examples are provided of integrable quantum field theories whose
(diagonal) two-particle S-matrix amplitudes are Hermitian analytic but not Real
analytic. It is also shown that Hermitian analyticity is consistent with the
bootstrap equations and that it ensures the equivalence between the notion of
unitarity in the quantum group approach to factorised S-matrices and the
genuine unitarity of the S-matrix.
