Shimano Positron Indexed 5 Speed Derailleur. Operated by a non indexed shifter with a steel wire instead of cable. Very cool stuff from the 70's.
Yes, those "frankenstein" systems work perfectly when new. I see a lot of them since I started a full-time job in a bicycle workshop (even new bikes, like a combo shifters-Altus, front derailleur-Alivio, rear derailleur-Deore XT), but only when new. When something starts to wear (like shifter for example) and everything won't even try to work. A week or so ago there was a guy who had the same problem as you - whole bike on Deore from group M591, except of the rear derailleur - an SLX M622. Everything from year circa 2012. The derailleur wouldn't want to work with the Deore shifter. New cable? Nah. New outer casing? Nah. New shifter? Nah. The only solution for the problem was to order a new rear derailleur from the group Deore M591.
On the other hand I have a client who has a complete set of Deore XT from about 2010. The whole set made +- 38000 miles of heavy beating, and it fails to shift properly cause the shifter is used up. But still the problem is that you have to "overpush" when shifting up, not that the derailleur shifts like it has a mind of it's own skipping two or more gears.