Note to self:
To use a smarthost you have to authenticate to with exim4, edit /etc/exim4/passwd.client
# password file used when the local exim is authenticating to a remote # host as a client. # # see exim4_passwd_client(5) for more documentation # # Example: ### target.mail.server.example:login:password *:myusername:myawfullylongpassword
Here I added a username and password for all target mail servers (*) because this host sends everything through the smarthost.
If yours does not, you have to be more specific (like the example).
Surprisingly, there is no need to reload or restart exim4 after modifying the passwd.client file.