as the reverse lights in both the manual transmission switch, and the neutral safety switch both have 12v present in run and accessory positions on one leg of the reverse light portion of the switch, [the neutral safety switch uses the two outboard wires for the reverse lights, while the standard transmission only has two wires for the switch, both used for the reverse lights] the connection to complete the circuit is when the shifter activates the contacts of either switch in the reverse position, completing the circuit to the reverse lights.
unplug the switch connector and verify you have 12v on one wire with the key on in the run and accessory position.
if no 12v present in either wire, your problem is upstream of the switch.
if 12v is present in either wire, jumper the two wires together with a piece of the appropriate gauge wire that will fit snugly into the connector sockets, and if the reverse lights work, your problem lies with the switch. if not, inspect the wiring from the switch to the reverse lights for issues.
if the switch seems suspect, it could be bad, or not adjusted correctly to activate it when the transmission is put in reverse.
if the lights stay on continuously, 12v are being constantly applied, which could indicate a broken reverse light switch, or a wiring issue in the connector running to the reverse lights, usually contained with the rear light harness, somehow allowing a cross feed of 12v to the reverse lights at all times.
i hope i have explained this so you can follow along.
if not, hopefully, others can chime in with better explanations.
please keep us updated on your progress.
