It must be related to the location. I just opened one of the sample forms in design mode, and then used the Publish command. I said I wanted to publish it to a web server and entered the URL where the file would be stored. As soon as I clicked Next, I got the prompt for that rogue domain again (as before, this different domain is not part of the publish location or part of our travel intranet, as far as I know). Even though I have full access to this server, it will not let me save it directly. My login credentials do not work for this strange domain. If I click cancel from the login prompt, it just takes me back to the screen where I type the desired publish location.
So instead I chose to publish it to a network drive, and selected our development server (which is a mapped network drive). Then, on the following screen in Infopath, I entered the full URL of the form on the intranet as the access location for users. Then it said it had successfully published. I manually uploaded the form from the development server to production, and get the exact same error. Is there any way to tell the form to automatically ignore any login prompts?