  We prove that any real, vacuum gravitational perturbation of a 4-dimensional
vacuum pp-wave space-time can be locally expressed, modulo gauge
transformations, as the real part of a Hertz/Debye potential, where the scalar
potential satisfies the wave equation. We discuss relations with complex
perturbations, complex space-times, non-linear structures, and real spaces with
split (ultra-hyperbolic/Kleinian) signature. Motivated by generalized notions
of parallel spinors, we also discuss generalizations of the result to other
space-times.
