Had same issue in IIS web Adaptor. Added server (B) to server (A) host file and fixed the problem.
It looks like the Web adaptor won't accept FQDs, so our DNS didn't know where to find it.
David,
Imagine my relief finding this in my google search! I also have set up Tomcat without a home directory, not knowing it would be an issue! I am going to create a home directory for Tomcat, if that fixes the issue you've resolved a week of headaches!
Edit: It did fix the issue. Thanks, David, I hope you know you helped someone three years after this post