Ok, sorry forgot to put in specific connections. I think the route would look like this:
LHR-BKK-(via SYD)-CHC-overland-AKL-(via SCL)-LIM-overland-CUZ-AQP-overland-VVI-(via SCL)-EZE-overland-IGR-AEP-overland-EZE-MAD-CAI-overland-AMM-(via MAD assuming RJ is member)-RAK-LHR.
I think this means that I only have 4 flight segments within S. America which should be ok. Feel free to correct me!