Another idea that occurs (depending on the order you need to travel to your destinations in) is to visit the Middle East straight after Asia, although that still has some issues:

SYD-AYQ-PER-NRT-HKG-PEK,HKG-DEL,BOM-BKK,SIN-HKG-DXB,THR-LHR-CAI-BCN-IST,MAD-JNB-MRU-JNB-VFA,NBO-LHR-EZE-SCL-IPC-SCL-SYD

Again, this has 23 segments as well as the Asia caveat, so fine-tuning is required. You would also only be able to transit LHR on the way from NBO to EZE. No extra Middle East flights are needed though.