I just bought a used bike with a broken 10 speed XT rear derailleur. Would a NEW Shimano XTR RD-M9000 SGS 11-Speed Shadow+ MTB Rear Derailleur Long Cage ( 182006158222 ) be compatible with 10 speed XT shifter & 10 speed Shimano XTR 11-36t Titanium Cassette that I have?
I can say from experience this does not work well at all.
Although as mentioned above the chain does run through the shifting is dreadful. I tried it with a 10 speed shifter but gears would always have a spot where they would jump.
Just get a 10 speed rear derailleur. You will be glad you did.
Will it work? Probably. The range of movement of the mech wont be an issue and the chain will run through it, so I cant see why it wouldnt work. This article is quite helpful: http://velonews.competitor.com/2015/03/bikes-and-tech/technical-faq/technical-faq-10-11-speed-drivetrain-component-compatibility_362722
Shimano will tell you 11spd components wont work with 10spd though. If I were you I'd just get a 10spd rear mech.
I can say for experience this does work perfectly (I used SLX 10s shifters + XTR M9000 SGS 11s rear derailleur + XT M771 10s cassete (with a 42t cog) + SRAM X01 11s chain). In the video below you can check a Shimano Zee 10s shifter working perfect with a Shimano XT M8000 11s rear derailleur.
Another report that it works perfectly: