In your own domain, this will work without a glitch. But not in the other domains!

When working in your local domain, the new name mysite is registered in your DNS.  But visitors from other domains who you give the url mysite.mydomain.tld will see that their http queries do not get resolved, and your web browser might suddenly try to search mysite on Google.

The issue is that Sharepoint employs Alternate Access Mapping, and will rewrite headers to the correct mapping, which is offcourse not correct in your multiple-domain environment.

The following Microsoft TechNet article explains how to adjust your access mappings.  Make sure you make a mapping with the fully qualified domain name, and new http requests will resolve correctly.