Your first step should be to check on if you will be allowed to conduct business on the type of visa you get and how to go about it. There are shoppers on the forum who have traveled internationally for mystery shopping. SteveSoCal comes to mind. He should be able to offer some great advice.
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This was my experience: I've traveled to Australia and mystery shopped while there last year. BARE Intl has shops. I also did an online only mystery shop for a MSC that is in the US while I was out there. I would suggest sticking with companies that offer shops in the US and Australia. Maybe you'd need a work visa to do work with any MSC overseas, but I was already registered with BARE and didn't have any problems.
Try picking up a few airport shops while you're stateside too.
Can't give you any advice about doing mystery shopping in Australia, but I can say that I hope you have as wonderful a time on your visit as I've had on my visits. Sydney's a great city. With luck you'll be able to travel beyond Sydney too. Enjoy!
If you are traveling on a Tourist Visa to Australia, then you can not work while in Australia. If they catch you, the fine is up to $10,000 and you will be deported. [www.findlaw.com.au] Try to find Post-Security shops at the airports you will be leaving / arriving / transiting in the US. You MIGHT also be able to do shops in the duty free shops post-security in Australia, but check carefully before you try that.
Shopping Southeast Pennsylvania, Delaware above the canal, and southwestern NJ since 2008