Exim4 with multiple domains

[unimatrix]

Ubuntu/Debian systems allow for a quick and easy configuration of Exim4 using the following command:

dpkg-reconfigure exim4-config

I have a very basic problem. I want to use multiple domains (or as Exim calls it "system mail names"):


As we can see it only allows you to enter one domain name:



Is Exim really so limited that it can only handle one domain?
How can I send and receive emails from more than one domain?
Exim forces me to choose one, which works, but when I try to send emails from another domain the receiver will still receive it from the main domain. GMail for example will display that email was sent "via <main domain>".