It would seem to me that the moral of this story is that when doing an ex-EU, if you know your positioning flights, it can be worth using a travel agent.
Just curious, are the TA fares the sum of the parts?
If yes, what I said in
https://www.flyertalk.com/forum/30584528-post132.html is wrong, it's not airlines trying to stop travellers avoiding married segment fares.