An existing page has a collection of modules on it. Then customer created a second page and added Existing Modules on it from the first page. When exporting that page as an XML template, any page created using the template doesn't have the shared modules but instead contains a copy of the modules.
Ideally, we should be able to create a template (Export as XML) and then create pages which contain those shared modules. It doesn't matter if we Include Content when generating the template or not. It is expected the newly created page to have the modules just like the page that the template was created from. The initial page's modules all show the message "This module does not belong to the page. Please, move to its master page to change the module", but the page created from the template does not.
This is the expected behavior since templates are intended to be shared not only on the same Portal but between Portals and other installations. Therefore, the content should be present even if the source module cannot be referenced.
If this module is meant to be present in most pages in the Portal then using the Add module to all pages feature from the module settings would serve this purpose, as these modules are added to all new pages by default, and the users can manually remove them from pages where they do not wish to have them; nevertheless, this might not be the most suitable option if you only want it for a few pages.
Another more complex option is to add an iframe or another way of referencing said module from your skin, and then use that skin for the pages.