Going to wp-admin causes logout

I’m having a mild inconvenience with login. The problem occurs when I’m logged in.

(1) Whenever I go to http://www.example.com/wp-admin, everything works fine: I’m already logged in, so I’m redirected me to the dashboard.

(2) However, whenever I go to http://example.com/wp-admin (without the www), it logs me out and have to log in again.

I’m familiar already with this page in the Codex. I’ve checked the siteurl value in the wp_options table. My home and site URL files are correct (http://www.example.com). It’s not a browser-specific issue: the issue occurs in every browser. I’ve deleted and regenerated my .htaccess file. I’ve replaced wp-login.php with the file from a fresh copy of WordPress.

Clearly I can avoid the problem by just making sure I always go to http://www.example.com/wp-admin instead of http://example.com/wp-admin, but I’d like to fix it if possible. Alternatively, if this is a default behaviour, it’d be good to know this so that I can attempt an .htaccess modification.