In AlF3-based glasses (CLAP), variations of oxygen as small as 0.2 wt% are observed to have order of magnitude effects upon glass stability, to the extent that monophase glasses can be formed only at normal quenching rates when the oxygen concentration is 1.1 ± 0.3 wt%. Glass compositions within this concentration band were prepared and studied by extended X-ray absorption fine structure and Raman spectroscopy. These data and oxygen dependence of other properties, some of which have already been reported, are used to propose a structural model based upon the coexistence of chemically perturbed regions, of the order of 1 nm in size.