Well then, here they are!
Unliving: When this model goes out of action, roll a d6. If a 1 is rolled, place a Resurrection die in place of the model with the face matching the number of wounds the model had (6 if the Model had 7 or more wounds). At the end of each turn, you may pay any amount of karma to reduce a resurrection die by the same amount. When the resurrection die reaches zero, place the model it replaced back on the field. This Model cannot have Necromancy.
Necromancy: When an allied model with Unliving goes out of action within 12" of this model, they place a resurrection die on a 2 or less. At the end of each turn, you may reduce the number on a resurrection die within 12" of this model by 1. This model cannot have Unliving.
I got inspired by the rules of the hunter edge, Smoke grenade, and how you can use dice to indicate a more persistent effect. That, plus the new way Karma works, Led me to this idea where instead of possibly giving you back a model immediately with wounds, you instead get an uninjured model, if you succeed the roll and have the karma to "pay off" the wounds. Necromancy also has more interaction with this new "resurrection" dice, now not only improving the chances of Unliving working, but also effectively giving you 1 free karma each turn, but only for ticking down a single Resurrection die.
I also included the exclusivity clause on both of them because this way, people won't feel pressured into picking both for every model in their outfit or otherwise try to loop Unliving Necromancers, instead giving them the potential to go in different directions with this synergy. Do you wanna make your boss and underboss unliving while your minions have necromancy, potentially representing an eldritch cult worshipping and bringing back truly monstrous beasts? Or do you put Necromancy onto a single model while the rest have unliving, making it more of a shambling zombie army with a lich at the helm? Given the whole "pay karma to get your models back", combined with the new 5 turn limit, perhaps the roll on Unliving's out of action success chance should change, something like 2 or less base, 3 or less with the Necromancy buff, but that would require playtesting that I definitely haven't done yet.