“You can see yourself in a pearl. It’s not like any other stone,” Mohamed al-Mahmood says with a smile. Then, he produces a showstopper from a red-velvet-lined box: The distorted yet highly sought-after form is known as a baroque pearl. It’s set in a necklace that will sell for $3.5 million. Mahmood is standing in the inner sanctum of his family’s jewelry store in Bahrain, a 274-square-mile island nation in the Persian Gulf that has been synonymous with natural pearls for centuries. It’s also a destination that American travelers can now reach easily for the first time. In October, Gulf Air inaugurated a direct flight to the country, departing three times weekly from New York’s JFK. Robb Report Travel Master Cari Gray of Gray & Co. says that many of her clients opt for commercial carriers, usually a Middle Eastern airline, for long-haul trips here, and swap to a private plane for the last leg...