GavinW: Well, that's not ENTIRELY true. Think of a sideband connection as a total "clean slate".
You can absolutely send Basic Auth via a sideband connection, you just need to manually construct and send the "Authorization" header to the server in the transaction -- in other words, don't wait for the server to 401, just go ahead and send the header it wants, formatted properly. The Authorization header is basically the username and password joined with a colon, then base64 encoded. See here:
http://en.wikipedia.org/wiki/Basic_access_authentication
Pretty simple on the BigIP:
set username "user"
set password "pass"
HTTP::header insert Authorization "Basic [b64encode "${username}:${password}]"
... or in the case of sideband, the header would just be another line of text you send out the socket. Cheers!