I don't think the "shared framework for coexistence" is multiculturalism because that isn't well defined.  I think what keeps people civil is the legal system.  A legal system based on enlightenment principles of freedom and equality naturally supports multiple ideologies and cultures...to the extent that these cultures aren't colliding.  For example, it isn't legal to stone people in the USA even if some cultures do it, but if people want to smoke a hookah that's their business.

OTOH a lot of polarization is created by intelligence agencies as a means of soft power to change a nation's government.  It isn't usually organic or the result of culture clashes.

When polarization is the result of fundamental irreconcilable differences then I think that is Casus separatio and I favor independence movements.