mod_auth_sspi

The effects of SSPIOfferBasic On are:

Internet Explorer + nonsense domain:
no difference, still automatically authenticated

Internet Explorer + correct domain:
no difference, still automatically authenticated

Firefox + nonsense domain:
no difference if you give the correct name and password, still authenticated
If you give an incorrect name and password, then basic auth box appears, but whatever you type in authorization is refused

Firefox + correct domain:
no difference if you give the correct name and password, still authenticated
If you give an incorrect name and password, then basic auth box appears. If you give the correct username+password then you are authenticated.

In summary: SSPIDomain is obeyed by the basic authentication box, but not by the automated authentication.

I don't know about trust relationships or other particular details of how the domains are setup.