I'm Australian and I visit Australia twice a year so no big deal for me. I'm switching to DTV later this year so I'll be doing that from Australia as well
Depends on the "untaxed income". All income I earn in Australia is already subject to taxation, with the exception of pension which is exempt tax in Thailand. I have to declare any income earned in other countries on my Australian Tax return. However, if this is already taxed in another country, I am allowed credits against Australian Tax. I used to own an apartment in Germany on which I paid tax in Germany. I declared the income on my Australian Tax return, which made it "liable" to tax in Australia, but as I'd already paid in Germany, this offset anything I would have had to pay in Australia.
Doesn't quite work that way in all situations. Because of my situation, although I'm a dual tax resident, I am deemed to be tax resident solely of Australia. In this case Australia is my taxation authority, and Thailand is out of the loop, except as I say for money earned in Thailand