who do you think would win in a real battle

some info about the Su-37
http://www.sci.fi/~fta/Su-27.htm

and here
http://www.sci.fi/~fta/Su-37-2.htm

russian mambo jambo vs american stealth high tech
I would say, if the Su-37 can get a lock on, then the F-22 is history. But as long as the F-22 remains stealth, the Su-37 is still very maneuverable enough and equipped with newest modern jammers to avoid those missiles(somehow)

so:
high maneuverability vs overal excellent specs + stealth technology

I hope this is not too off topic...