I opened Australian bank accounts with ANZ and CBA back in 2011/12. I emailed them explaining what I wanted and eventually made appointments to go and see them at the appropriate branches when I was next passing through Sydney. There was a bit of to-ing and fro-ing about it all, but in the end, we managed to work our way through the AML/ID process.

I know that that's not particularly helpful at the moment, but I suspect that there will be a trans-Tasman travel bubble some time toward the end of this year.

in Australia I run transaction accounts - debit card only as neither ANZ nor CBA will issue credit cards to a non-Australian resident - as well as both term and currency deposits. I investigated opening sharebroking accounts, but they too are limited to Australian residents.

I made sure that I signed up for online banking at the same time, and can happily shuffle various currencies in and out of Australia, to and from various places, with no issues, as well as setting up term deposits.

As far as the Australian term deposits go, as they mature they're moved off into the Russell Investments Australian Government Bond ETF (RGB.AX). Same currency, same asset class, better liquidity, better credit rating, and better yield.

Just remember that what you're trying to do is not an every-day run-of-the-mill thing for most bank staff - the "secret" is to find someone competent - which is often easier said than done in the world of financial services.