KORZOK, India—It was dusk when the herdsmen reached their Himalayan village bearing ominous news: They had spotted dozens of camouflage-clad Chinese soldiers inside territory India considers its own.

Indian security forces poured in, beginning a face-off last month that grew to involve more than 1,000 troops on each side at an altitude of roughly 15,000 feet, according to Indian officials, making it the biggest border confrontation between the two nations in decades.

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