You slightly misremember that (Robbers cave study). The effect is even more powerful: all it took was the seperation into two groups and being made aware of the existence of the other group for them to develop an identity and name for their respective group.
The specific proposal I would make for the US is what I call the People's Congress, replacing the House of Representatives, leaving the Senate and executive roles as-is (other than unrelated electoral reforms like RCV).
As a check against abuse of power, or nominating someone unfit, every randomly-selected candidate can be voted out, albeit requiring a very high margin (maybe 66%: "the People's Veto"). Note that this position would be a mandatory form of public service, not unlike jury duty. Terms would be two years, and representatives would cycle in and out on a regular, rolling basis.
A realistic proposal? Not even slightly. In the current climate we can barely get basics like mail-in voting right. (Maybe it's something that be done as proof-of-concept at the scale of an eccentric municipality?) Nonetheless: my hunch is that having a branch of government, possibly even the entire legislative body, being statistically comprised of We The People in a way that is impossible to game, would generally be less corrupt, and more responsive to the needs and preferences of constituents.
The thing is, from a game-theoretic standpoint, tribal loyalty is a Schelling focus [0]: your calculation about what alliance(s) to invest in, are inherently influenced by your calculations of the calculations of others. What defines the tribal boundary can be utterly non-sensical (from the objective viewpoint of a proverbial Martian), yet still be a rational choice from the perspective of each individual participant.
I'm also reminded of the Netflix b-movie "Circle" [1]: a group of fifty people are forced to vote every two minutes on which of them will be killed (if they refuse, someone is killed at random). At first, they decide to buy time to think, by starting with older people who are closer to the grave; but the older crowd bands together in self-defense, creating a two-party system of old-vs-young. As the headcount shrinks, new rifts and coalitions emerge; some organic, some arbitrary. It's a brilliant thought experiment despite being terrible as a film. :)
[0] https://en.wikipedia.org/wiki/Focal_point_(game_theory)
[1] https://en.wikipedia.org/wiki/Circle_(2015_film)