We explore the space of consistent three-particle couplings in Z(2)-symmetric two-dimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [2] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [1]. The space of allowed couplings that we carve out is rich in features, some of which we can link to amplitudes in integrable theories with a Z(2) symmetry, e.g., the three-state Potts and tricritical Ising field theories. Along a specific line our maximal coupling agrees with that of a new exact S-matrix that corresponds to an elliptic deformation of the supersymmetric Sine-Gordon model which preserves unitarity and solves the Yang-Baxter equation.