Something that is important to understand in this context is that the US is about 75% white while South Africa is about 20% white. This means that enforcing white ethnic power in South Africa was going to be far more challenging. The solution that was found was that certain areas were to be declared “black homelands”. These areas were supposed to be made independent (though South Africa never allowed them real independence) and black people were to be citizens of these “black homelands” and not of South Africa, leaving South Africa a white state. As such black people were not citizens of South Africa and were not legally able to live outside of these “homelands” (although those restrictions were more in paper than in reality).
To summarize: South Africa had a black majority while the US had a white one. As such enforcing white power in South Africa required much more extreme measures.
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