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If you have wire strands broken in the casing, or something of very thin gauge, your situation is possible. A voltmeter has very high resistance, so when you take the voltage at the masthead, it makes sense to get the same voltage you would get at the battery terminals, because the circuit is not loaded. DC circuits suffer from voltage drop; when they are loaded (the masthead light is on) , the voltage decreases, because a battery is not a perfect voltage source, and the wires have resistance. As the circuit heats up, the problem gets worse.
I would think that if the masthead light worked before, and stopped working, that you have some area that does not conduct well under load, which explains why the multimeter is throwing you a curveball! There is not a lot you can do, in a practical fashion, to find these in a continuous run of wire. I would suggest looking in places where the wires bend, entering the mast, and perhaps elsewhere, and carefully inspecting these sections, then replacing as necessary.
Good luck finding your ghost. This is part of why marine grade wire is a good thing!