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Some people say yes, some people say no.
From the literally hundreds of posts that I've read, I've come to these conclusions.
37s will fit on stock wheels and on the Raptor.
Now, having said that, I've read some folks reporting minor rubbing at full turns.
Even more people reported serious rubbing and damage off-road. I guess if depends on how much compression you subject your wheels/tires to, but a lot of anecdotal posts that say "warning".
So, most folks suggest doing a mid perch mod. The back seems to be ok. The front needs a bit of help.
Even with the mid perch mod, there are some reports of rubbing, and some of bumping off road. Here's where the conversation turns to the pinch welds, cutting fenders and fender liners, glass fenders, etc. I usually tune out when the conversation goes this direction. Not interested in modding my truck like this just to fit 37s.
So, besides the mid perch, new wheels with 0 offset may help. They'll push the tires out about 1.33" on each side. Some folks say this is good even off road, others say no way. At this point there might be a variance in tire width (12.5 vs 13.5) but few folks include their full tire size when commenting.
Soooooo, after all that, I decided to get new wheels. I wanted black anyway, and the offset could help. I'm getting mid perch done, and hoping 37x12.5x17 Goodyear Wrangler MT/Rs will fit mostly for street driving and some (very) mild off-roading. If I run into issues, I think some minor fender liner trimming might be required. I think trying to do this on the stock wheels will be harder.
It seems people with similar setups report different results. You'd think there is some variance between trucks. Some folks swap for another brand of 35" tire and report rubbing, because the BFG 35" tire is like 34.5". I find it hard to believe .5" can make that much of a difference, but apparently it does.
This is my take on this whole tire/wheel thing. Expect some rubbing on turns with those 37s on stock wheels, and some fender damage if you go off-roading. Consider mid perch as a step towards mitigating that, but that is not a solution. You'll need new wheels with a different offset, and possibly new fenders if you want to go seriously off road. For street driving only you "might" get away with it but the tight turns are going to be a problem unless you push those tires out a bit and raise the front a bit.
Tire wisdom from the newb after having gone cross eyed reading "yes it fits" and "no it doesn't" 100 times. I just said "screw it" and bought my 37s, and let the rubbing happen if it will. I'll deal with it when and if it does. YMMV.