BUENOS AIRES—A 43-year-old governor has unexpectedly become the face of President Mauricio Macri’s efforts to dismantle the legacy of Peronism, the movement that has dominated Argentine politics for the past seven decades.

María Eugenia Vidal, governor of Buenos Aires, Argentina’s biggest and most populated province, has taken on Peronist public-sector unions and purged corrupt security agencies in her first year and a half in office. Along the way, polls show, she has become the country’s most popular politician.

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