Fine, said I, I'll redirect /index.php to /index.shtml which is the actual file I wanted. But then I discovered that that redirected all accesses to / to /index.shtml which is not what I wanted (I prefer leaving the index page off the end if at all possible, to make for a cleaner URL).
The problem was, it seems, because when you request a directory (e.g. http://www.tetrap.com/drwho/) the web server searches a list of possible file names that your index file could be. Evidently, index.php was earlier on the list than index.shtml.
Sticking this in my .htaccess file fixed it:
DirectoryIndex index.html index.shtml
I guess it'll also speed up accesses to directories marginally, as the server will only be looking for those two filenames.