Maybe this would work.

lon-rio//sao-lax-las-sfo-yvr-jfk-yto-jfk-nrt-hkg-sel-hkg-bom-bkk//sin-syd-akl-syd-jnb-cpt-lon

Lots of short hops and only one addl segment in Asia. You would need to get from Bangkok to Singapore, which is reasonably cheap, and from Singapore to and back from Kuala Lumpur, which is also cheap. Also, you need to get to Sao Paolo from Rio, which isn't too much. Finally, I am presuming you can go to Goa from Bombay. I honestly don't know intra-Indian prices.

If you can take Goa out, you will make your life easier.