Nematic elastomers with locked-in anisotropy direction exhibit semisoft elastic response characterized by a plateau in the stress-strain curve in which stress does not change with strain. We calculate the global phase diagram for a minimal model, which is equivalent to one describing a nematic in crossed electric and magnetic fields, and show that semisoft behavior is associated with a broken symmetry biaxial phase and that it persists well into the supercritical regime. We also consider generalizations beyond the minimal model, particularly the neo-classical model introduced by Warner, Bladon, and Terentjev, and find similar results. Our work underscores the fact that semisoft response is fundamentally a nonlinear phenomenon and that experiments on samples at zero external stress provide no information about it.