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 (Pm (3) over barm) to rhombohedral (R (3) over barc) 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.