Eg suppose you have one system that lets you verify 'this person has X dollars in their bank account' and another system that lets you verify 'this person has a passport of Honduras' and another system that lets you verify 'this person has a passport of Germany', then whether the authors of these three systems ever intended to or not, you can prove a statement like 'this person has a prime number amount of dollars and has a passport from either Honduras or Germany'.
I see the big application not in building a union. For that you'd want something like Off-The-Record messaging probably? See https://en.wikipedia.org/wiki/Off-the-record_messaging
Where I see the big application is in compliance, especially implementing know-your-customer rules, while preserving privacy. So with a system outlined as above, a bank can store a proof that the customer comes from one of the approved countries (ie not North Korea or Russia etc) without having to store an actual copy of the customer's passport or ever even learning where the customer is from.
As you mentioned, for this to work you need to have an 'anchor' to the real world. What ZKP gives you is a way to weave a net between these anchors.
As I understand it, you can do arbitrary computations on https responses and prove that you didn't tamper with the response or the computation.