ORLANDO, Fla.—In his bid to become the vice presidential nominee of the Libertarian Party, former Massachusetts Gov. Bill Weld is facing a trial by fire in the quirky internal politics of his adopted party.

Mr. Weld, who was twice elected as a Republican in a heavily-Democratic state during the 1990s, unexpectedly re-emerged on the national political scene this month, announcing his intention to seek the vice presidency of the U.S. as a Libertarian.

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