The tobacco refugees scattered to Florida, the Caribbean, Central America, even the Canary Islands off the coast of Africa, searching for soil and climactic conditions like Cuba’s, the perfect combination for producing rich, slow-burning leaves. They recreated their old labels.

Today, there are two called Partagás, two called La Gloria Cubana, two called Montecristo, two called Romeo y Julieta, and two called Hoyo de Monterrey—one made for the U.S. market, the other produced in Cuba to be sold everywhere else. It’s the same duality that reigns in the rum industry: Bacardi produces the expat version of Havana Club, while the Cuban government has partnered with Pernod Ricard to distill its Havana Club; the two sides have been feuding in court for years over control of the trademark.

Almost all the cigar refugees would in time sell their brand rights, and they eventually made their way into the hands of Scandinavian Tobacco Group A/S and the U.K.’s Imperial Brands Plc. (Imperial is aggressively playing both sides of the trade, having also formed a partnership with Cubatabaco to distribute all its cigar exports.)

The Quesadas kept the business in the family, cultivating a 155-acre farm in the Dominican Republic and also sourcing leaves for its blends from Honduras, Nicaragua, Mexico, Ecuador, Indonesia, the U.S. and Peru. Some of the blends, in fact, have elements of the old Cuban flavor. In the 1960s, someone managed to smuggle a supply of seeds off the island, and the company today grows five varieties of Cuban tobacco.

“A little envelope with grams of seed will grow a whole country,” Quesada says.

That Cuban tobacco grown in Cuba soil is a superior raw material to that found anywhere else on Earth is of little debate among the cognoscenti. There’s just something about the orange-hued clay of the Pinar del Rio valley that runs along the island’s western tip. Where there’s some disagreement is whether that raw-material edge always translates into a better finished product.

For Giuseppe Ruo, the cigar sommelier at London’s Wellesley hotel, the answer is an unequivocal yes. He finds Cuban cigars so far superior that he refuses to stock anything else in the hotel’s walk-in humidor, the U.K.’s largest. But Jorge Luis Armenteros, who trains cigar experts at Princeton-based Tobacconist University, sees things a bit as Quesada does. The communist government just doesn’t have the entrepreneurial spirit, he says, to keep pace with the exiled artisans and newcomers who’ve taken up the craft in surrounding countries.

Today, Fonseca is one of four brands that Quesada Cigars sells. Each brand contains multiple lines; each line contains multiple shapes and sizes. They consistently make their way onto lists of the world’s best smokes compiled by Cigar Aficionado and Robb Report.

Quesada never doubted he’d come face to face with his old Cuban nemesis at some point, and he figures there’s at least one surefire way to ensure he can profit from such a scenario: striking an accord over the Fonseca name, with Havana agreeing to have his company distribute the Cuban versions in the U.S.—or possibly allowing him to make his own cigars with Cuban tobacco. Both sides would profit instead of squandering time and money in court. (There are many other elements to Quesada’s plan that he didn’t want to discuss publicly.) Habanos SA, a unit of Cubatabaco, didn’t respond to requests for comment.

The incentive to avoid a drawn-out legal battle will be great for both sides, says Frank Herrera, a lawyer who knows how messy such disputes can be. One of his clients, Miami’s Guantanamera Cigars Co., got sucked into a more than decade-long court battle after the Cuban government launched its own product of the same name and claimed the American producer was misleading people into believing its cigar was from the island—Guantanamera is a popular Cuban folk song. The case is ongoing.