No real easy way to do this as a "RTW" staying in the southern hemisphere, particularly as each ocean segment is served by different alliances...
PER-JNB is done by *A, SAA.
SYD-JNB is done by QANTAS.
AKL-SCL by LAN.
JHB-EZE by Malaysian Airlines.
Probably the best bet, is to buy a AKL-JNB ticket with a stopover (or 2) in Perth. The AKL-JNB fares are on sale at the moment, ~$2200 NZD. I think QANTAS codeshare on PER-JNB. It should still cost around $2300 NZD including the stopover. You'll have to buy another PER-AKL ticket, and you'll have another PER-syd/mel-AKL segment that you can use in the future.
Alternatively, two tickets, PER-JNB and PER-AKL. Although a
*A ticket RTW "might" be cheaper...
Also, if you do this correctly and credit to AAdvantage, then for 10k points + ~$100 booking & taxes you can do PER-syd/mel-AKL oneway. (or 17.5k in Business). I think US Airways have a similar deal for *A, though it's supposedly pretty hard to get PER-AKL availability on NZ.