In order to get this working I had to check out the lastest version of the Ruby bundle:
cd ~/Library/Application\ Support/TextMate/Bundles rm -rf Ruby.tmbundle svn co http://macromates.com/svn/Bundles/trunk/Bundles/Ruby.tmbundle
I suspect the problem was an old copy of the bundle (not sure how it got there). Deleting it probably would have been enough.
There still seems to be a way to go before this will become useful. In a brand new file I found that hitting Option-Escape would allow me to complete basic strings like
Struct. But on trying it on anything remotely complex would yield only a tool-tip containing the words, "Parse error".