Green-hyperbolic operators are partial differential operators on globally hyperbolic spacetimes that admit advanced and retarded Green operators, whether or not they are actually hyperbolic in the traditional sense. They have proven very useful in the study of QFT on curved spacetimes. In this talk I describe a generalisation of Green hyperbolicity to allow nonlocal perturbations, which have various applications, including modelling the effects of noncommutative spacetimes, and certain aspects of measurement theory for quantum fields. Under suitable conditions we prove the existence of Green operators and also determine the potential obstructions to their existence, namely whether there are smooth solutions to the homogeneous equation with past or future-compact support.