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A motel staff member said she recognised the family as soon as they walked in yesterday asking to book three rooms - but they denied being the same group of travellers.
Meanwhile, the British High Commissioner to New Zealand says she "sadly" cannot revoke the passports of the unruly tourists.
And British media have reported that the travellers are well-known for their anti-social behaviour while living at a site near Lutterworth, Leicestershire.
One nearby resident said the family had caused trouble there for many years.
"They copulate, defecate and urinate. It is never-ending."
Yesterday, 26-year-old Tina Maria Cash pleaded guilty to stealing energy drinks, a rope and sunglasses from an Auckland service station on two separate occasions.
On January 3, Cash allegedly returned to the Caltex with her children and stole more Red Bull, rope and sunglasses valued at $50.
Her lawyer told the court she does not remember the incident, but accepts the theft was caught on CCTV video.
She has been convicted and ordered to pay $55 in reparation.
The motelier said the group of travellers also tried to stay there on Monday night but she refused their request because of their rude behaviour, so yesterday requested the family pay a $360 bond for the rooms to protect the business and they agreed.
She said she didn't have any complaints about their behaviour from any of the guests, some of whom also recognised them.
Each of the rooms were in a satisfactory condition when they checked out.
Asked for her opinion on their behaviour while in New Zealand, she said it was disgusting and not how people should act.
"I can't tell British visitors what to do. I think that would be overreaching my powers."
Clarke also urged Kiwis to remember that this family doesn't represent the majority of British tourists.
"New Zealand is the most beautiful country and so many British people want it come and visit here each year and they have an amazing time and this is a tiny, tiny minority."
"The vast majority of British tourists who come here have an extraordinary time and behave really well and this is the exception."
"It's unfortunate but I wouldn't want to draw any conclusions because as I say, hundreds and thousands of people come each year and behave really well and go home with wonderful memories of New Zealand, and the same is true in the other direction."
Meanwhile, Cash and her family have been identified by British newspaper The Sun as having a reputation for anti-social behaviour in their native Britain.
Cash had lived with her family on a traveller site near Lutterworth, Leicestershire.
"We've had sheep chased and killed by their dogs, they've used our land to take drugs and farming equipment has been going missing.
"It's the younger ones. They just do whatever they want to do."
There had also been problems with thefts from local stores and anti-social behaviour near the local Catholic church: "They have no respect for the people who are trying to pray."
One resident believed that Cash had lived in the area until 18 months ago, while others had seen her there more recently.
Travellers have also been banned from a local pub after starting a fight on New Year's Eve.
Honorary Consul-General for Ireland in New Zealand Niamh McMahon has warned Kiwis not to label the group as Irish.
"It's a tarnishing of Irish People here who are by association labelled, that this is the behaviour of all Irish people," McMahon told Stuff.