γ softness in atomic nuclei is investigated in the framework of energy density functionals. By mapping constrained microscopic energy surfaces for a set of representative nonaxial medium-heavy and heavy nuclei to a Hamiltonian of the proton-neutron interacting boson model (IBM-2) containing up to three-body interactions, low-lying collective spectra and transition rates are calculated. Observables are analyzed that distinguish between the two limiting geometrical pictures of nonaxial nuclei: the rigid-triaxial rotor and the γ-unstable rotor. It is shown that neither of these pictures is realized in actual nuclei, and that a microscopic description leads to results that are almost exactly in between the two geometrical limits. This finding points to the optimal choice of the IBM Hamiltonian for γ-soft nuclei.