BUENOS AIRES—In 2010, Rui Saito, a Japanese tango dancer, and her Korean partner were competing in the prestigious citywide competition known as El Metropolitano when Ms. Saito says an Argentine woman in the audience came up to them and said she didn't appreciate their presence. "'What are foreigners doing dancing tango?'" Ms. Saito said the woman told her. "'You don't know tango.'" Later, Ms. Saito says the woman heckled them as they danced.

This year, things got more bleak for foreigners at the Metropolitano. Organizers...