In case anyone else has this problem, here’s the work-around I came up with:
The authentication window is coming up because the login page is getting styles from the CSS folder inside wp-admin. So I copied the contents of login.css and colors-fresh.css and embedded them in the login page inside a style tag. I also had to put a copy of logo-login.gif in the images folder above where wordpress resides. (This image is referenced in one of the styles.)
Frankly, I’m baffled why no one else seems to have this problem (I searched and searched). It seems to be common advice to put password protection on the wp-admin folder.
