So I know that Islam allows tribalism, but do tribal ties carry any weight in Islam? Do one's tribesmen have any sort of special rights towards you? Does one have any obligations towards his or her tribe? Can one start his own tribe or clan?

Those close to us have rights on us. Our neighbors have rights on us more than those who are far. Your relatives and tribesmen also have rights. What Islam fought is injustice and racism. I oppress someone because of my tribe. I reject the truth because of my tribe. I take people’s rights and give to my tribe. This is the tribalism Islam fought. Otherwise being good to your tribe, serving your tribe and bringing prosperity to your tribe us good. You can start your own tribe (if it’s possible—not sure if this is possible nowadays).