The Klingon Neutral Zone was established by the Kobayashi Maru at the beginning of Star Trek II. I suppose it could've been the RNZ, but that would've made Klingons hanging around a lot stranger. Or maybe not. The Romulans and Federation are both forbidden from the Neutral Zone, so the Klingons fly through with impunity picking fights.
Or, just spitballing here, the Klingons and the UFP agreed to set-up Neutral Zone of their own after the thing with the Organians. The less contact between the two forces, the less often their ships and weapons get all hot, glowey and unusable. And the whole Romulan Neutral Zone seemed to work for the most part, so it was a workable model to use.