SILVA JARDIM, Brazil—After school and on weekends, Carlos Leandro Peixoto de Abril sells ice cream made by his grandmother from a stoop alongside the family's cinder-block home.

Instead of Brazilian reais, though, the 11-year-old prefers payment in capivaris—a local currency emblazoned with the face of a giant rodent. Bills in hand, Carlos then heads to a local grocer and buys ingredients, at a special discount, for another batch of grandma's goods.

The...