[Omatako]

Why would a 12v panel put out 18v?

Not to me. Once you connected them in series, it should be 24v. We need an expert here, I get a headache.

But there is clearly something more to it, if you test the panel and it shows 12v (or more likely something in the 13s). But I don't see how, from that diagram.

Don't need to be an expert. The diagram clearly shows the panel drawing from one battery. How can it be 24v if it is only drawing from one battery?

And the job of the regulator is to set the output voltage to the batteries at an acceptable level. The OP said that the solar array was unregulated.