on the automotive relay, like any other relay, there are two leads for the coil, and then there are three more control leads - one is common, the other one is NO, normally open, and NC - normally closed.

you would tap the light LED wires to draw power to coil leads, when the reverse LED is energized, the coil is activating, and NO leads are closed, and the NC leads are broken open... you connect desired signal to control leads and feed it to your trigger wire.