OK, I assess the entire career, live albums included, but rereleases and compilations excluded. The idea is to simplify the question by eliminating pairs of albums (yes, I call them albums) that are approximately equivalent in quality. You eventually wind up with a set of records upon which you can make a decision. Of course, this all my opinion. I unapologetically love both bands.
These pairings are roughly equivalent in my view.
Achtung Baby
Murmur
The Joshua Tree
Reckoning
War
Automatic for the People
All That You Can't Leave Behind
Fables of the Reconstruction
Zooropa
Accelerate
The Unforgettable Fire
Live at the Olympia
How To Dismantle an Atomic Bomb
Green
New Adventures in Hi-Fi
Hasta La Vista Baby
No Line on the Horizon
Reveal
Rattle and Hum
REM Live
Pop
Out of Time
SO, with those equivalent selected discographies out of the way, here's what's left to decide who's better
REM: Life's Rich Pageant, Document, Monster, Up, Around the Sun
U2: Boy, Under a Blood Red Sky, October
Since Monster's not great, and I hope to never hear Up or Around the Sun ever again, I will follow U2.