I'm not quite sure. Chromium (the open-source end of Chrome) is very much a cathedral in the sense of product direction, but is developed in the open. I can't come up with something that has the opposite pairing though (it'd be kind of strange).

I'm all in favor of some pithy term for "source released under an open license but not developed in full view of the public."

How about "write-closed" vs. "write-open"? Or more simply, "read-only open source".

Fork-only might be more appropriate, since you're quite free to write to your copy.

Shared Source is nice, or half-open source.

Or, "we take all the benefits, free workforce and goodwill of free software while retaining all control and money".

The first thing I thought of was "Shared Source", the licensing program for Windows source code.

Most 'cathedral' products are developed at least somewhat in the open, and often do take external contributions.