My experience has been the only drink I can drink with any cigar is water. I have also experimented with Wines, Brandies, Cognacs, Rums, and Whiskeys. I am intrigued with the Anejo Tequilla pairings and will soon try that.
My experience has me liking rye whiskey, Speyside and Highland scotches and white or gold rums with lighter bodied cigars. Non Cubans would be Sticks like Rocky Patel 55, or Romeo y Juliet Bully. For the Cuban varieties with these same drinks The first to come to mind is El Rey Del Mundo Choix Supreme, and I was surprised by this one, Hoyo de Monterey Epicure #2.
I could go through the bourbons and the rums and all the other liquors as I have tried with cigars here too or I can cut things short and say Cigar flavor strength and liquor flavor strength can pair well and enhance the flavor of both! However for me most of the enjoyment comes from the experimenting and trying different pairings.
So smoke em if you got em and thanks to whomever started this.