In a large-scale, preregistered experiment on informal political communication, we algorithmically matched participants, varying two dimensions: 1) the degree of incidental similarity on nonpolitical features; and 2) their stance agreement on a contentious political topic. Matched participants were first shown a computer-generated social media profile of their match highlighting all the shared nonpolitical features; then, they read a short, personal, but argumentative, essay written by their match about the reduction of inequality via redistribution of wealth by the government. We show that support for redistribution increased and polarization decreased for participants with both mild and strong views, regardless of their political leaning. We further show that feeling close to the match is associated with an 86% increase in the probability of assimilation of political views. Our analysis also uncovers an asymmetry: Interacting with someone with opposite views greatly reduced feelings of closeness; however, interacting with someone with consistent views only moderately increased them. By extending previous work about the effects of incidental similarity and shared identity on affect into the domain of political opinion change, our results bear real-world implications for the (re)-design of social media platforms. Because many people prefer to keep politics outside of their social networks, encouraging cross-cutting political communication based on nonpolitical commonalities is a potential solution for fostering consensus on potentially divisive and partisan topics.