In the path integral picture of QFT, we have to integrate over all paths connecting two states. What happens when we only integrate over paths that don't violate causality? Since the paths that are closest to the classical paths contribute the most, I would expect that this might be pretty close to the path integral that does include these acausal paths.
Here is a rough sketch. The causal paths are entirely timelike, so they never move faster than the speed of light.
