Overproof Rum Drinks - A rum with a higher alcohol content than regular rum. Rum is produced wherever sugar cane grows. Many counÂtries, such as the United States, South Africa, and even Russia, produce rum, but it is only the Caribbean Islands that produce rum in quantities sufficient for worldwide export. The islands in the Caribbean each produce a distinctive type of rum, the result of the base material used, the method of distillation, and the length of maturation. Generally, the islands where the Spanish language is spoken, such as Puerto Rico, produce light, dry-tasting rums. The English speaking Caribbean islands produce dark, heavy-tasting rums.