In an effort to clear tension with Barack Obama, Philippine President Rodrigo Duterte approached him during a dinner at a regional summit in Laos last month, two days after he had openly criticized the U.S. president.

It made things worse. Mr. Duterte didn’t feel Mr. Obama treated him as an equal in the encounter, says a Philippine official present, because Mr. Obama said a follow-up would come from White House staff, not himself. The next day, Mr. Duterte boycotted a group meeting with Mr. Obama and Southeast Asian leaders.

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