Well, they do. The rules actually do not even mention the word "surreply" so apparently they literally don't exist. If you want to file yet another brief after the specifically allowed batches, you do have to file a "motion for leave to file sur-reply," or at least that's what other Second District cases have done. There's no rule allowing such a motion. This is generally after the case is set for argument, but it was fully briefed before then, or it would not have been set for argument.
All gnat-fucking aside, if the court can simply skip further briefing and decide the case on the papers, it is for all reasonable purposes "fully briefed," even if the rules permit, but do not require, a party to submit additional papers if they are lucky or fast enough to get it in before the court decides it or sets it for oral argument.
After it has done that, of course, any further attempt to submit papers would have to be by leave of court, and then they would be submitted specifically named something like "motion for leave to file sur-reply," even though you rarely see the alleged "motion" actually argued in depth, and it generally simply consists of the actual brief that the court would accept if it chose to do so.
A "reply" after full briefing as permitted in the rules is just a class of the general object of sur-replies, and is customarily called that. I seriously doubt there is actually any case law addressing whether or not something is specifically a "sur-reply" except in the context of when it is necessary to file leave for one, because it simply doesn't matter.
A naming convention does not determine what something actually is, though, nor does an awkward styling make something impermissible that is otherwise permitted by the rules. They're trying to turn an argument about what it is customary to entitle something into one about what it actually is by the accepted definition of the terms, even though it's absolutely a nugatory dispute.
It's pretty telling how far these guys want to dive into the weeds far, far away from anything resembling the substance of the briefs, though.
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