CIENFUEGOS, Cuba—Fidel Castro and Hugo Chávez proclaimed a decade ago that they presided over a single country, combining Cuba’s educated workforce with Venezuela’s oil wealth to challenge U.S. power across Latin America.

Now Mr. Castro is gone, three years after Mr. Chávez’s death, and the union between the two countries, while still strong on paper, is withering away fast.

Daily shipments of more than 100,000 barrels of subsidized Venezuelan oil, the lifeblood of Cuba’s economy, have dropped by more than half since 2013, according to oil traders and Cuban refinery workers. In November, Cuba had to buy oil on the open market for the first time in 12 years as Venezuela’s output plummets.

Meanwhile, thousands of Cuban doctors who toiled in Venezuelan shantytowns to pay off the oil deliveries are quietly returning home, scaling back an important vestige of the popular social programs Mr. Chávez left to his now embattled successor, Nicolás Maduro. The air bridge between the two Caribbean countries is also dissolving: Cuba’s flagship airline, Cubana de Aviación, stopped regular flights to Caracas earlier this year. Charters from Caracas to Havana have scaled back too as demand slumped.

On the surface, leaders in both countries swear to an ironclad coupling that detractors mockingly call Cubazuela.