ANZ says it dismissed seven staff who deleted customer email addresses to prevent them giving negative feedback, which may have impacted the employees' incentive payments.
In a statement, the bank said about a year ago it found some staff were manipulating customer records and those involved were disciplined, with seven losing their jobs.
It said it randomly asked some customers to respond to an email survey about their experiences, but some surveys weren't being sent.
First Union represents frontline bank employees, including 900 ANZ staff.
Its general secretary, Dennis Maga, said banks' bonus culture was to blame because it made staff desperate to avoid negative feedback.
ANZ announced in August last year it was removing sales targets for frontline staff.
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Robertson said it was not his preference to hold a royal commission, but also indicated that any further revelations of bad behaviour by banks would make the pressure hard to resist.