I have the same problem and I have tried nearly everything I found on the net. It's definitely related to having permalinks turned on in the wordpress. However, I have put a new .htaccess file in the subdirectory with:
and the problem still exists. Even if I completely delete the wordpress .htaccess file the problem still exists.
I have also tried some other suggested solutions such as ErrorDocument 401 "Unauthorized access" and ErrorDocument 404 "Unauthorized access" and Redirect 301 /mysubdirectory http://www.mydomain.com/mysubdirectory/index.html in various locations all to no avail.
Can someone please offer another solution? The only way I can fix it is to turn permalinks off but we need them to be on.