From the code in the original question:
request.AuthenticationType = "urn:federation:authentication:windows";
That particular line didn't appear to have any effect. Also, the hope that no HomeRealm need be specified was also in vain. It turns out that you must, in fact, specify the HomeRealm. After searching google for some time, I found many statements hinting that the Active Directory trust was the same as trusting the ADFS server itself. So as a wild shot in the dark I tried putting in the URI of the ADFS server, and, much to my pleasure and surprise, this worked.
So, to be more clear, if you go to the web.config of the relying party you should have something set up similar to this:
<issuerNameRegistry type="System.IdentityModel.Tokens.ValidatingIssuerNameRegistry, System.IdentityModel.Tokens.ValidatingIssuerNameRegistry">
<authority name="http://testadfs.test.com/adfs/services/trust">
<keys>
<add thumbprint="XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX" />
</keys>
<validIssuers>
<add name="http://testadfs.test.com/adfs/services/trust" />
</validIssuers>
</authority>
</issuerNameRegistry>
I ended up setting the value of request.HomeRealm to the value that you see there in the validIssuers and authority elements. This gave me the desired effect of causing ADFS to automatically select the Active Directory claims provider trust.
I should be clear that my ADFS server is NOT the domain controller, and yet this still works.