  Green-hyperbolic operators - partial differential operators on globally
hyperbolic spacetimes that (together with their formal duals) possess advanced
and retarded Green operators - play an important role in many areas of
mathematical physics. Here, we study modifications of Green-hyperbolic
operators by the addition of a possibly nonlocal operator acting within a
compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal'
generalised Green operators. Assuming the modification depends holomorphically
on a parameter, conditions are given under which $K$-nonlocal Green operators
exist for all parameter values, with the possible exception of a discrete set.
The exceptional points occur precisely where the modified operator admits
nontrivial smooth homogeneous solutions that have past- or future-compact
support. Fredholm theory is used to relate the dimensions of these spaces to
those corresponding to the formal dual operator, switching the roles of future
and past. The $K$-nonlocal Green operators are shown to depend holomorphically
on the parameter in the topology of bounded convergence on maps between
suitable Sobolev spaces, or between suitable spaces of smooth functions. An
application to the LU factorisation of systems of equations is described.
