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The San Juan Islands, an archipelago off the coast of Washington State long cherished for its remoteness and simplicity, is now the hottest luxury real-estate market in the country.

The priciest 10% of the area’s real-estate market rose faster over the past year than any other U.S. county, according to Realtor.com. The median home value in San Juan County is now $444,300, up from $377,600 in 2014, according to real estate…