Brent Peak and Lisa Everett typically fall on different sides of political issues. He leads a local chapter of a progressive advocacy group. She’s a GOP district chair. Yet when the two met this past spring on the sidelines of dueling protests, their willingness to talk with one another led to finding some common ground – on the white-hot topic of immigration.

That day, Mr. Peak’s group gathered outside the local offices of the U.S. representative for Arizona’s 8th Congressional District. Several carried signs protesting cuts by the Department of Government Efficiency. Ms. Everett joined a counterprotest, sporting a Trump T-shirt. The two met through a conversation about keeping their demonstrations peaceful. They kept up the discussion over breakfast.

The Democrat and Republican have si

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