In the setting of integrability in AdS/CFT, in the typical approach to the AdS_5 x S^5 string a light-cone gauge is fixed, breaking Lorentz invariance on the worldsheet. One attempt to avoid the need for this goes under the name of Pohlmeyer reduction. Recently an S-matrix was conjectured to describe the scattering of solitons in this theory. Now as I will explain in my talk, the S-matrix of an integrable quantum field theory together with its dispersion relation are enough to find its finite volume spectrum exactly, through the so-called thermodynamic Bethe ansatz. I will work this out for an S-matrix and dispersion that interpolate between the standard light-cone gauge fixed superstring and this conjectured Pohlmeyer S-matrix, by analogy to a simpler model. Viewed as a deformation of the light-cone gauge fixed superstring TBA, this story is very similar to deforming the XXX spin chain to the XXZ one, which I will concretely discuss. I will finish by emphasizing important differences to this simple toy model, and discuss surprises in the so-called Y-system associated to the TBA equations.