BUKAVU, Democratic Republic of Congo—A dozen artisanal coffee aficionados from around the world hovered their noses above cups of steaming grounds and inhaled deeply. Then they tasted the coffee, swishing or chewing it, spit out the leftovers and used calculators on their smartphones to tally a final score for each sample of beans.

“I thought this was so great. It was so fleshy,” head judge Beth Ann Caspersen, who lives in Rhode Island, said about the highest-rated brew in one round of the competition, limited to coffee grown...