One inconsistant with Enterprise itself. Recall "In a Mirror, Darkly", where a Constitution-class starship smashed through 22nd century human and Andorian ships like tin foil?
But In a mirror darkly was set in an alternate timeline -- the Evil-Cochrane obviously wasn't the one seen in First Contact.
And included information about "our" Archer and Hoshi in the database (and I'm going by what was said, not illegible screen graphics) Or the finale, which definitively put Enterprise in the past of The Next Generation.
Lets make the assumption that the Defiant in In a Mirror Darkly was the same defiant from the same universe as TOS. Lets also throw in the on-screen shot of First Contact (2061), which was the pre-FC accepted date for TOS, and ties in nicely with the young looking Cochrane (in FC it's a bit of a stretch to think he was only 33).
In that case, perhaps the fork was further back? While Voyager in Futures End existed in a universe that was more advanced that ours back in 1996 (perhaps due to 29th century technology, but there was a model of a DY-100 ship in Rain's office). That world was one which could have easilly led in to a TOS universe which has a youthful 31 yearold Cochrane performing the first warp flight in 2061.
There's no doubt that Archer and the Enterprise NX1 existed in the same timeline that a fat Riker encountered phase cloaking technology.
Voyager also made several references to events from the First Contact movie, and also crossed over with TNG, DS9 and referenced TOS at several points - putting them all in the same future history (countless in-universe continuity issues aside)
We know from the alternative universe from Mirror Mirror onwards, but especially in DS9, that there seems to be something that binds the specific Spock-Goatee and our universe, more so than repeated crossings (how likely is it that all the main cast from ENT, TOS, DS9 and Tuvok, would come to exist - the chances are uncalculable. Literally. A single sperm being slightly off course at some point in the previous 400 years for any of the ancestors of any of the characters we saw would have prevented them from existing)
The alternate universe storylines in TOS/DS9/ENT (not neccersarilly TNG though) are the biggest proof of a higher power (God, Q, Wesley) existing across the multiverse.