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?
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: