With the negative battery cable disconnected, and the positive cable connected, I'm measuring positive 12.6V battery voltage between the negative battery cable and the negative terminal on the battery.
Yes, the multimeter works, and I didn't have the probes reversed. I also measured 6 ohms resistance between the disconnected negative battery cable, and the fusible link (with both positive cables removed).

I think there must be a short someplace, but I haven't been able to find it.

The vehicle was running normally until the alternator stopped charging the battery. I installed a new alternator, then noticed this situation.

Where should I look, and what should I check?

 

If you want to be sure, use a test light instead of a multimeter. You're probably getting transient voltage in the low milliamp range. Nothing that would power anything, but it would show up on DMM.

 

I don't have a test light, so I used a turn signal bulb, a couple wires with alligator clips, and multimeter to measure current. I connected one side of the light bulb to the negative battery cable, and the other side to the multimeter positive probe, multimeter negative probe connected to the battery negative terminal.

When I first connect it, bulb lights up brightly, then gets dimmer. Multimeter shows 330mA when first connected, then fades to 250mA and stays steady. Same fading behavior from light bulb if the multimeter is not in the circuit.

So that's around 3W of drain. Seems odd. Is that enough to be worried about?

 

It is neither surprising nor alarming that your multimeter is reading volts between the negative battery cable and the negative battery terminal. There are systems that are always active, even when the truck is not running and the key is removed from the ignition switch. The BCU is one example. And if you listen carefully in a quiet place the ABS module makes noise.

The question should be about amps, not volts. Switch your multimeter to read amps and see how much current is being drawn with all systems off.

 

With the multimeter connected between the negative battery cable and the negative battery terminal, it's measuring 350mA.

 

I don't remember for certain but that sounds in the normal range.

 

OK, this makes me feel so relived, I was recently out on a forest service campsite, it was super quiet and I could hear a faint buzzing coming from each wheel, which I figured had to be related to the braking system (or I was going crazy). Is this a cause for concern or just another facet of the Discovery 2 “character”?

 

This is totally normal, since all the devices connected to the battery have no infinite internal resistance to their ground points (leakage of electronic components) and the voltmeter has very high internal resistance (my multimeter has 10 MOhm).

This is totally normal too. When you switch to amps, the multimeter has very low internal resistance. When measuring you connect the battery again through the ammeter.