To clarify:
E.g. I am using Strange Delusions to e.g. give a character the Agency faction to attach Dynamite to that character during the same phase.
Does the attachment stay attached and usable after the phase?
Strange Delusions:
Attach to a character or support card. Any Phase: Exhaust Strange Delusions to give attached card a faction affiliation of your choice until the end of the phase.
Dynamite:
Attach to an Agency character. Action: Discard Dynamite from play to wound all characters committed to the same story as attached character.
I've also been wondering: Does Strange Delusions replace the faction or does it add a faction to a card?