Mikey and Graham...
The new 13-pin plug obviously contains all the connection wires.
It is then possible for the battery charging connection, which in a detached 13 pin plug is directly connected to the +ve terminal of the caravan battery, to short out to another terminal if the wire becomes detached for some reason at the back of the 13 pin plug.
Say in Grahams case this wire has become loose in the connections at the back of the plug and is touching the feed wire terminal to the caravan rear lights.. then you have seemingly mysterious rear lights that are illuminated.
Under the old 12N and 12S system the rear light, brake light, indicator circuits could not get a +ve supply until the plug was plugged into the socket on the tow car.
It is even possible for water contamination to allow power to flow across terminals.... thus my recommendation to try using a water dispersing spray.