London — It is one of the world’s most exclusive clubs, known over the years as the Syndicate, the Central Selling Organization and the Diamond Trading Company.

For more than a century, De Beers has sold most of its rough diamonds to a select number of customers, a list that reads like a who’s who of the opaque gem-trading world. Tiffany & Co, Graff Diamonds and Signet Jewelers all own subsidiaries in this group, guaranteeing a steady supply of gems with the pedigree of being vetted by De Beers...