Are there any non-touristy tribal experiences in South Africa?

If you're expecting colourfully dressed, exotic-looking tribal peoples that you can visit and for it not to feel commercial, then no that's difficult to find. An experience like that that you’re paying for will, by definition, be somewhat inauthentic, if not exploitative.

However, if you want to meet people of different tribes but not necessarily dressed in an exotic manner and dancing around for you then that's possible everywhere in South Africa on almost every street.

For something more in-between, and I suspect, more what you're after then I'd try heading to the Kalahari regions in the north. It's a stunning semi-desert region (lots of interesting wildlife and a real wilderness feel) and is the home of South Africa's small San Bushmen community. These were some of the original inhabitants of southern Africa and some do live a very traditional lifestyle (but well away from tourists).