Hi Jennifer.
You've got it exactly right. Whilst at the NEC, we talked to the guys on the Thetford stand, as we have had exactly the same problem. They were extremely helpful, and knew all about this problem.
It's a black algae which builds up in the tank, and their recommendation is to put Milton in the tank which will clean all this gunge out.
You'll probably have to flush it a couple of times to get rid of what's already in there completely, but then add a capful to each full tank, and the problem will not return.
Whatever you do DO NOT use bleach, parazone, or any type of domestic toilet cleaner as this will destroy the seals.
Domestic products are not good for plastic toilets for a number of reasons, and the guys were most emphatic that we didn't go down that route.