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Effective Field Theories (EFTs) for Goldstone Boson scattering at a low order allow the computation of near--threshold observables in terms of a few coefficients arranged by a counting. As a matter of principle they should make sense up to an energy scale $E\sim 4πF$ but the expansion in powers of momentum violates exact elastic unitarity and renders the derivative expansion unreliable at much lower energies. If new--physics deviations from the Standard Model are found and encoded in low-energy coefficients, perhaps at the LHC, it will be profitable to extend the reach of the EFT to regimes where partial waves are saturating unitarity. The methods known in hadron physics as "Unitarized Chiral Perturbation Theory" extend the EFT up to its nominal reach or up to the first new physics resonance or structure (if found below that energy reach) in the partial wave amplitude, but they usually have unknown uncertainties. We recapitulate our analysis of the systematic theory uncertainties of the well known Inverse Amplitude Method (IAM).