We report on diffuse neutron scattering experiments providing evidence for the presence of random strains in the quantum spin-ice candidate Pr2Zr2O7 . Since Pr3 + is a non-Kramers ion, the strain deeply modifies the picture of Ising magnetic moments governing the low-temperature properties of this material. It is shown that the derived strain distribution accounts for the temperature dependence of the specific heat and of the spin-excitation spectra. Taking advantage of mean-field and spin-dynamics simulations, we argue that the randomness in Pr2Zr2O7 promotes a new state of matter, which is disordered yet characterized by short-range antiferroquadrupolar correlations, and from which emerge spin-ice-like excitations. Thus, this study gives an original research route in the field of quantum spin ice.