  We present a version of relative locality based on the geometry of twistor
space. This can also be thought of as a new kind of deformation of twistor
theory based on the construction of a bundle of twistor spaces over momentum
space. Locality in space-time is emergent and is deformed in a precise way when
a connection on that bundle is non-flat. This gives a precise and controlled
meaning to Penrose's hypothesis that quantum gravity effects will deform
twistor space in such a way as to maintain causality and relativistic
invariance while weakening the notion that interactions take place at points in
spacetime.
