As a New Zealand citizen entering Australia, I am automatically granted a Special Category Visa that allows me to live and work in Australia indefinitely.
When I enter Australia I have the option to use the Smart Gate system which automatically grants a Special Category Visa on entry. This is paperless, ie my passport is not marked with a visa stamp.
When proving to potential employers that I do indeed have the legal right to work in Australia, all I can present is my passport, which is unmarked and as clean as the day it was issued.
As it seems to be done electronically these days, is there any web site or Immigration Office I can visit that will let me get some sort of certificate that proves my visa and therefore my legal right to apply for work?