JOHANNESBURG—Dorothée Mauchle sets out three meals a day for a pair of cacophonous companions most South Africans go out of their way to avoid.

“Haddie haddie haddie!” she called on a recent afternoon from her home’s patio in this city’s northern suburbs. Moments later, amid a thunder of dissonant screeching and wing flapping, two turkey-sized birds descended from a nearby oak.

The...