This is actually a semi-common problem in the Western world. See also: Spain and Britain. We've seem to hit a point where populations in some countries are split evenly and there is no unifying political will.
The Athenian solution to this, if I recall, is ostracization. Every year citizens vote by writing the name of a single citizen. The citizen with the most votes, provided a minimum bar of 6,000 had been met, was banished for ten years.
I think this would have the effect of preventing polarizing figures. If you were so polarizing a large portion of the population hated you...
I could imagine adapting the practice for modern sensibilities. Create a resort run by federal money out on the beach of some tropical island. Free food, drinks, games. It could double as a tourist attraction so there would be people to hang out with. If you win the ostracization vote you get a five year vacation on the state and all of your previous positions or titles are withdrawn.
