Definitions of interoperability surround us, but all the attention in the world to definitions make very little difference in the end.
I have a theory that people will believe two systems are interoperable when they can observe that the systems work together in simple ways with little to no effort, and in complex ways with some effort. In neither case does that effort require substantial coordination between multiple parties (to achieve interoperability at the technical level).
This is a theory rather than a definition, and needs to be tested. It's based on interoperability between applications I use on a regular basis. Here are some of the applications and/or technologies I use, and some answers to whether or not I think they are interoperable.
Is a spreadsheet interoperable with a database? Most are. You can readily use a spreadsheet to query and update a database.
Is a word process interoperable with the PDF format? Not quite. You can easily generate PDFs, but you cannot read PDFs or update them without the full blown edition of Acrobat.
Is a word process interoperable with XML formats? Again, not quite. I can read and write a specific XML format (or perhaps even multiple depending on my word processor), but not generalizably so. It takes a bit of work to get useful XML out of some word processing content.
Is social media interoperable with my Word Processor? No. Can I make it work? Of course I can, but I don't think that my writing software to make it work fits within the definition of interoperability.
Is my e-mail interoperable with my blog? Yes. It readily supports simple stuff (notifications and postings), but not complex interactions without a good bit of work.
Is my e-mail interoperable with my tablet and smartphone? Yes. Frustratingly though, I cannot do everything on my tablet that I could on my computer. More than enough to make it frustrating to not be able to do more...
Is a patient registration system interoperable with an EHR or departmental system? Most require a bit of work to perform this "simple" and even "designed" capability. Is that interoperability? Most users would probably say no.
What is your theory of interoperability? How would you test it?