We don't believe this is possible. The justification for this belief is nothing less and nothing more than experimental observation. We have never observed a process where an effect comes before its cause, so we simply reason inductively to establish a postulate that the preferred order of events in physical processes is always the same, for any observer.
Indeed, it is this postulate that leads us to the conclusion that there is no physical inertial frame that travels at greater than $c$ relative to any other; as long as this is true, then relative motion cannot make event order observer dependent. On the other hand, if there were such a frame, one can always find a physical process whose events are reversed in time by the supraluminal boost.
See my answer here for more details.
See also the Einstein Tolman Paradox, also called the Tachyonic Antitelephone. This is almost exactly analogous to your scenario.