Interesting topic. Ironically, I have worked for both Ringling Brothers and Barnum & Bailey Circus and currently with Cirque du Soleil. (I am currently a SM with Cirque.) Both companies do have stage managers but with RBBBC you will see it listed as Production Manager and Associate Production Manager. These positions do the 'stage managing' per say but during the show they will call curtain cues mainly and wonder a lot about what might be going wrong on the floor. The board ops (lx and sound) do their own cues. With Ringling the PM and APM deal with contracts, disputes, local publicity, show notes, artist interviews and, the biggie, the animal walk that happens in each city.
With Cirque it is handling a lot of the backstage areas, calling the show, show reports, dealing with the tour publicist, assisting in getting the weekly rehearsal schedule ready, etc. There will possibly be 3 positions on tour being the General Stage Manager, Backstage Manager and Junior Backstage Manager.
Both companies are a lot of fun to work for but like every other company, they have their ups and downs.
If you have further questions, please ask.