Who would even run such an organization? the EU? God help them.
The EU may actually be a better bet than anyone else. One advantage of a semi-independent bureaucracy is you aren't actually beholden to political whim, so the professionals can get on with what needs doing. Some of the more useful disability rights legislation has come out of the EU rather than Westminster, Eurocontrol is leading on air traffic control modernization, ESA is a major force in space exploration, the EU has taken the lead on greenhouse gas emissions trading and so on.
This isn't to say there aren't simultaneously problems, but a transnational EU Shadow Unit is actually a reasonable possibility.
Huh. My impression was that they spent a majority of their time arguing about trivial bullshit. But perhaps this is just the European Parliament, and I've gotten the wrong impression by getting my info on Europe from US news sources.