[exim] pretend a different domain

Hi,

with several domains declared as local domains, users can
send and receive messages with sender addresses the domain
part of which is any of the local domains. Unfortunately,
when looking at the headers, the recipient can always see
from which domain the message was actually sent because of
headers like Received: and Sender:, and the envelope information.

Is there a way to make it so that the recipient of a message
sent from <user@???> cannot figure out that the message
was actually sent from <user@???> when the host exim
runs on has a primary hostname like mx.domain-a.net?


I´m having users who should have an email account for two domains,
and when sending test messages, I can always clearly see in the
headers where the messages actually came from.

Do I need to set up another email server for this?