Complete non-powered circuit by 12v trigger source?

BOB is certainly correct!! If the DC supply for the remote receiver is different from the 12 volt relay latching scheme, you might need to add a diode to the +12 out from the remote, maybe. Remotely switching it off will be a bit more complex. IT could be done with a single channel remote, a momentary press to switch on and a press and hold to switch off.
It could also be done by using an alternate-action relay.