So I've been playing around with this quite a bit, with the online tool, SABRE and with the AA desk. ex-CAI different airlines have published different fares. It appears QR and AA have published fares in USD while the rest are publishing in EGP. Further, it appears that the fare selected is based on the "dominant" carrier, which seems to be the first overwater/long-haul carrier OR the first carrier. For me here's what I saw:

CAI-xDOH-DXB-xLHR-PMI-xMAD-DFW-GRU-JFK-xHKG-DXB - priced out at USD $6K+
but changing it to
CAI-xLHR-PMI-xMAD-DFW-GRU-JFK-xHKG-DXB (and ensuring that I only use BA, IB and CX for the over water segments) - priced this at $3442. (validating on CX)

If I used the AA flight from MAD to DFW - it prices using AA base fare (>$7k), if I use QR in the first segment it uses QR base fare (>$5k).

With these restrictions in mind - I was able to get the AA desk to price the itinerary at around $3,800 (validating/ticketing on AA) - higher YQ and taxes but perhaps worth it to deal with the AA RTW.

Good luck