Just inspired by some video's of someone ranting about side missions.
A scientific experiment in quantum entanglement causes a party member to be instantly blamed for something wrong in a passerby's life. Land on a planet, step out and security is blaming you for the recent cockroach infestation. Wander out into the desert and a passerby will accuse you of being in cahoots with their cheating husband and tell you to tell him they hope he enjoy's the surprise (booby trapped hunting traps) they left him. Even being fired into a black hole via an escape pod wont work as the other people on the pod will be blaming them for the fact they're currently falling into a black hole. Basically wherever they go and whatever they're doing there someone will be blaming them for anything between their pizza coming with no breadsticks to the destruction of entire planets, even the big bad of their current plot will blame them for their turning out how they did.
This naturally will get annoying afterawhile and prompt the party to look for a way to undo the effects of the experiment.