What foreign carriers can I use under the open skies agreement? I'm currently trying to book an academic international flight that I'm planning to get reimbursed by a US government-funded institution. The minimal rate with a US carrier is almost $1000, whereas the rate with certain foreign companies is about $600. 

I looked at this link and it has a provision that flight on foreign carriers is allowed if either there are no domestic options available (which is not the case for me) or (clause b) if it is allowed under the "open skies agreement", which seems to be a mess of conditions and exceptions which I don't completely understand. 

Is there a clear way or a clear heuristic to decide whether a given itinerary is allowed for reimbursed travel?
The best way to check this is to clarify with your institution's travel office (the one where you will submit the reimbursement).

However, my general understanding of the situation (From the days when I worked for the government) was that if there's a code-share between a US airline and the foreign airline you wish to use, they will generally be OK with that. The problem is if it's purely international, and there's no American airline that partners with them. Then you may run into some difficulty.