Madoka: "So tell me what you thought of Rebellion?"
But like to be slightly less glib... it depends on exactly what you are looking for. Do you mean in the context of two separate fanbases engaging in the same material in radically different ways? You have things like the Harry Potter franchise, which I'm sure has a sharp division between people that have solely watched the movies, people that have solely read the books, and both. Leila makes a good point about Star Wars. I'd toss Game of Thrones in there too. Honestly most large fandoms have this to some extent, because you splinter into movie continuities, TV continuities, comic continuities, and so on. Often times these groups are going to either exist completely independently and ignore the others, or outright clash, with intermixing, as it were, relatively rare. A kind of interesting example is early Sailor Moon fandom, which was splintered between people who saw the subs, back when that was much more difficult, and people that relied on the dub and dub knowledge, either because it's what they had, or what they preferred. That's died down because subs became much easier to get and you have a dub with reasonable fidelity now anyway, but that was an interesting thing back in the day. And even now you have a minor split between people that read the manga and people that watched and preferred the anime. (This is less of a split and more a preference, though, given how prolific the SM anime is.)
Now if we start talking about schisms and in-groups that emerge within fandoms, ooooh boy. Shipping does it a lot especially with big fandoms (Harry Potter with Hermione/Harry v. Hermione/Ron, I recall some DIgimon shipping drama with Kari and T.K., etc.). Heck, I'd say that if you get to a reasonable size, shipping schisms are basically inevitable with some really rare exceptions. There's also the schisms over sequels or spinoffs (Madoka is a good example of this, Star Trek is too as previously noted, I've seen some interesting arguments among Digimon fans about the topic, Dangan Ronpa, etc.) that can get unduly heated. There's debates and schisms over character's morality (Harry Potter, again, provides a good example with Snape, Dangan Ronpa provides a good example with Kokichi), the morality of sudden plot threads or elements and if they're justifiable, and in some games the very existence (let alone execution) of certain plot twists or game elements (again, Dangan Ronpa).
Like, you could really approach this question in a variety of ways. The way where you have isolated communities due to different source materials, separated communities due to receiving the material differently, or splinters and schisms among a formerly united fanbase that cause it to become permanently, or severely, divided. I think it shows how versatile media is there days, as well as how amorphous and flexible the idea of "fandoms" really is, that you can get a wide diversity of kind of answers to these sorts questions, alongside various examples that all excellently support their kind of answer.
