We found the solution.

First, a little bit about Edit Mode --

The tab strip above the Preview Frame appears to change based on the URL of the page in the Preview Pane having a querystring with its ID in it.  If you mouseover a URL in the Preview Frame, you see that it has an "id" param in the querystring.

Normal URL: /my/page/

URL when in the Preview Pane: /my/page/?id=123

This querystring param appears to be what triggers the tab strip to update.  (I suspect this is what the "idkeep" querystring param is doing -- it's telling the URL rewriter to keep the ID in the URL when it parses and corrects all the URLs in the page.  (I don't know where this update is actually triggered from -- either from onload on the frame or from some Javascript hook on the click event...)

Well, in this particular use case, we were generating some HTML remotely through a Web service.  So, links on the page that was loaded in the Preview Frame were being generated in another process, and that process had no idea we were in Edit Mode and that we should keep the "id" param on there (it had no idea the page into which it was dumping HTML had "idkeep" in its querystring).  So the link was the public link, with no id param.  Clicking on the link therefore did not update the tab strip.