Creating a virtual institute
I know a number of independent or semi-independent researchers who have set up a small company for themselves to affiliate with (e.g., the whimsically named Turing Ate My Hamster, Ltd.), for various different reasons.  

This is pretty easy to do and, I believe, achieves everything that you are looking for from a virtual institute.  It also has the advantage of not being fake in any way---just very small.