Looking for a pointer in the right direction ...
Is there a mechanism which allows you to configure SharePoint in such a way that:
if a user has been successfully authenticated within a SharePoint site that there is some kind of "authentication token" what can be passed or is available to 3rd party sites
or a way for 3rd party sites to "recognize" that the user is currently authenticated within a sharepoint environment
all 3rd party apps can be modified to accommodate whatever needs to be done but the constraint is: SharePoint may or may not be a hosted (by a separate service provider) and how the original authentication took place is irrelevant i.e. just need to know they authenticated ok, not how
EDIT scenario to help clarify: authenticated SP users require access to a 3rd party service provider for additional content. a "link" on their SP site redirects through to the 3rd party. the 3rd party needs to recognize the referrer (based on a collection of evidence supplied by the request) so that it need not challenge for a secondary authentication process.
one of the 3rd parties is me. the SP instances are many and varied and would be any one of my clients (which i don't offer support to, just provide a content service to).
so the attempt is to solve more of a general "community/ecosystem" problem.