I hadn't heard of the term. Here's a link to the Wikipedia article and pictures of the last San Diego ride.
Tweed Run - Wikipedia, the free encyclopedia
5th Annual San Diego Tweed Ride | Photo Gallery | UTSanDiego.com

It looks like a lot of ordinary people mostly riding ordinary bikes in conventional clothing with a few in genuine tweed. The only concern is the possibility of crashing a SWB recumbent in that large a crowd. I'm not sure I would do it on my Haluzak Horizon but would consider using a trike.