LinuxQuestions.org - Problem routing email from LAN host using Exim 4.43-1

I'm trying to route mail from a host on a private LAN to a valid external email address, and am getting this error:

Code:

    host myrealdomain.com [nnn.nnn.nnn.nnn]: 550-Verification failed for <nobody@myhost.localdomain>

    550-unrouteable mail domain "myhost.localdomain"

    550 Sender verify failed

The private host in question is (a web server) in a DMZ, and I have a similar problem with other hosts behind the firewall. The issue seems to be that the domain name is unrouteable, and therefore unverifiable. This is reasonable, as the domain name is meaningful only inside the firewall.

I would like to be able to route mail from root and from applications to external email. I would prefer a solution enabling me to name the sending address so that this can be properly validated by routing MTAs if possible. How might I achieve this, or otherwise what must I do?

Thanks for any suggestions!