Political polarization poses a variety of challenges for modern democracies. Entrenched disagreements on policy can prevent constructive discourse and compromise, and high levels of affective polarization threaten to undermine social cohesion and support for institutions. Finding ways to promote constructive discourse while maintaining free expression has proved a challenge for social media platforms, media outlets and policy makers alike. Here we develop a computational model -- based in psychology -- of online discourse and opinion dynamics under complex individual identities, which we use to assess the capacity of realistic interventions to prevent or reverse polarization. We show that changes to the range of acceptable opinions in a society -- i.e. the Overton window -- have a limited impact on polarization, and that attempts to ``optimize'' the Overton window can even trigger the onset of polarization. In contrast, interventions that shift attention towards under-discussed topics, or increase the costs of violating existing norms, are often effective at preventing polarization, but are less successful at reversing it. Most strikingly, increasing the salience of influential individuals, who model non-polarized discourse, can be highly effective at both preventing and reversing polarization. However we also find that once polarization has set in, even the most successful interventions result in latent extremism when identities are complex. Our work suggests that restricting speech by shrinking the range of acceptable discourse is an ineffective way to tackle polarization, whereas enforcement of existing norms, attention nudges and the presence of elites who model good behavior can be highly effective.

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