I should have asked what auth mechanisms you see in the relay EHLO response? A good way to test this (or any issue with SMTP communication in fact) is with the swaks tool.
-- Ian -- ## subscription configuration (requires account): ## https://lists.exim.org/mailman3/postorius/lists/exim-users.lists.exim.org/ ## unsubscribe (doesn't require an account): ## exim-users-unsubscr...@lists.exim.org ## Exim details at http://www.exim.org/ ## Please use the Wiki with this list - http://wiki.exim.org/