I was approached by a monster. A chain, from the Abyss- vaguely analogous to your demons, I think... he offered me the chance to change the past in return for passage into my world. To do that, I had to let him anchor himself in my body, and I had to feed him.

Unlike possession in your world, I remained well within control of myself as he fed. I made a choice to do what I did, to get them back.