After the initial WTF and some checking of my wiring I realised what was going on. You can see from this diagram that one end of the pickup is connected to the positive terminal. The other is disconnected and the wire connected to the earth is connected to the metal pickup cover.
Testing with a multimeter confirmed that there was no electrical connection between the case and the disconnected wire and yet when I plugged the guitar into my amp I could hear sound at a much lower volume than normal.
So if the pickup is not connected to earth how is the sound getting through? What is not shown on this diagram is that all three wires are in a single two core shielded cable. The close proximity of the wires meant that together they were acting like a capacitor. A capacitor blocks DC which is why the meter didn't detect any electrical connection, but it does let alternating current pass.
What it does illustrate is that sometimes you have to be careful of cable placement because it can have unintended consequences.