"Before we even get to that boiling point, these companies will find some middle ground,” Herrera says.

The most famous tussle is Cohiba v. Cohiba, over whether General Cigar Co. has the right to the U.S. trademark. Cubatabaco, which makes Cohibas of its own, has been trying to cancel that trademark in a case that’s been dragging on for 20 years. The backstory might be one reason for the Cubans’ persistence: Cohiba was created especially for Castro back in the 1960s.

For Quesada, the fight will be over his five Fonseca lines. They may not be not his best cigars—a critic recently labeled one of them a mere golf-course smoke, fit for those preoccupied with other things—but they account for a third of Quesada Cigars’ annual revenue.

The brand is legendary. Registered in 1907 by Francisco E. Fonseca, a Cuban-American with a Van Dyke beard whose visage appeared on the box, it eventually found its way to the exiled Sosa family in Miami, with whom the Quesadas teamed up before acquiring sole ownership in 1995. By then, Quesada had earned an undergraduate degree in the Dominican Republic, served an Army tour in Vietnam, gone to graduate school at Florida State University and returned to the family business.

To this day, he keeps a vintage Fonseca box in his office at the factory, where maps of Cuban and Dominican tobacco-growing regions cover the walls. He taps a thinning arsenal of prerevolution Cuban Fonsecas to enjoy at Christmas and other special occasions.

Quesada has made a nice life for himself and his family in the Dominican Republic, where one of his two daughters works with him in the business, along with a niece and nephew. He lives in a house he built in the La Rinconada neighborhood, with a pool and a man-cave stocked with a chaotic assortment of books, CDs, distilled beverages and, of course, copious amounts of cigars. He always wears a shirt with pockets, because he needs a place to stash his “children,” as he calls them. (If his wife asks, he smokes one or two a day, tops.)

The cigar industry has grown in Quesada’s lifetime and come under new pressures. Give him an opening, and he’ll rant about the U.S. Food and Drug Administration’s plan to mandate that cancer-warning labels be slapped on his carefully crafted and varnished wooden boxes.

But the process, for the most part, hasn’t changed since the days of the Spaniards in Cuba. At a palm-thatched barn about an hour from the border with Haiti, workers rely on time-honored methods to prepare recently-harvested tobacco. Quesada buries his nose in a pile of leaves. “Beautiful!” he exclaims, describing them as having just hint of sweetness.

Nearby, a man in a tank-top prepares a pile for delivery to the factory with the elaborateness of a religious ceremony. The leaves have to be just wet enough that they won’t crack during the journey, and he occasionally dips a shrub called azahar in a water basin and flicks the shrub at the tobacco. The tap-tap-tap of the droplets tell him when there’s sufficient moisture.

It is this dedication to the craft that Quesada swears you won’t find back in Cuba.