There are other solutions rather than using include(). For instance, if you don't need to execute PHP from the other server (which you should not do, unless you need your head testing 🙂 ), then you can use file_get_contents instead and echo() that.
That would have the effect of outputting the HTML from the other server, without parsing it for PHP code, which is probably more like what you want.
Other options include returning a structured response from the other server (XML, serialized form from PHP, JSON, etc) and then processing it to incorporate it into the design of your internal site.
It's also possible to use "screen scraping" - to parse the HTML from the other site and extract elements programmatically. If you have create the other application yourself, and have direct control over it (i.e. designers can't simply change the design without your say-so), this may be an option.
Personally I'd lean towards a server-server structured response, e.g. PHP serialised form (which is the simplest if both ends run PHP).
Mark