  We explore the space of consistent three-particle couplings in $\mathbb
Z_2$-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
[arXiv:1607.06110] 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 [arXiv:1607.06109].
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 $\mathbb Z_2$
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.
