i am sure this is a frequent topic, but in searches, i am getting results that are almost ten years old; and in any case, this is a topic where the answers and experiences change.
Up to recently, while i had a working AU debit card for all my daily Internet transactions, we have just been using the Agribank account that my VN wife had - and cash.
But the bank was sold to NAB, and NAB has closed all those accounts - and my debit card with it, and it is somewhat problematic getting an AU debit card again w/o a AU residential address or phone (although i may soon attempt to do that).
We then opened a new Visa account with Agribank (in my wife's name). That involved a bit of mucking around as you had to go to a bigger central branch to be able to apply and get the new card, and then there was more mucking around to get it authorised to use world-wide for Internet purchases.
So far, so so so; but the 'killer' was that Agribank, probably because of its roots in small rural communities are just not attuned to any modern banking needs - most critically, they can NOT tell you ANY details (except date/time/amount) for each transaction, ie name or details of who! This is exacerbated by the fact that you also cannot get a list of transactions (with no details mind) on your computer, or bank statements - only a list of transactions (and a balance on your phone. And finally, even though it is a different bank account, they will only let you use a ONE phone number for notifications - not a separate phone number, ie mine, to get notifications for that account.
Vietcom, where my smart SIL banks, so we thought that should be a good choice. On checking, yes, I can personally get a Visa debit card - BUT the restrictions on transferring money in or out are almost impossible. Virtually, the only thing i can do is put money in there from overseas. No cash deposits. NO transfers between bank accounts.
Now, that is if the account/card is in MY name. Thinking about it, if it is in my wife's name, as now, there probably is no problem; but i would have to check explicitly that it does NOT have all the various dysfunctionality that Agribank has.
So, long story, long: which bank would you suggest that has a large local footprint can i use that DOES do/have all those things?