Density measurements suggest that the incorporation of ZrF4 into the cubic ReO3-type structure of Sc1-xZrxF3+x is associated with the creation of anion interstitials. X-ray total scattering measurements are consistent with the conversion of corner-sharing octahedra to edge-sharing polyhedra as the solid solutions become richer in ZrF4. The cubic (Pm3̅m) to rhombohedral (R3̅c) cooperative octahedral tilting transition seen for ScF3 moves to a higher pressure as increasing amounts of zirconium are added, and it is eventually suppressed completely (x = 0.4 and 0.5) so that the cubic phase persists to high pressure until an amorphization occurs. All the samples studied (x = 0.0, 0.1, 0.2, 0.3, 0.4, and 0.5) display pressure-induced softening, and increasing the zirconium content leads to a higher zero-pressure bulk modulus. The incorporation of "excess fluoride" into ReO3-type fluorides is a powerful tool for suppressing the generally unwanted phase transitions seen when subjecting these materials to stress.