Immigration New Zealand served Andhoey a deportation notice on January 23 while the ship was berthed at Auckland's Viaduct, and he left the following day, said spokesman Steve Jones.
Mr Andhoey had 28 days to appeal the decision but chose to instead leave a day after receiving the notice, he said.
Mr Andhoey has said he planned to sail to McMurdo Sound to find out what happened to his three crew who died when their yacht Berserk sank last year.
Customs officials spotted the yacht in international waters on Thursday night and made numerous attempts to hail it but there was no response.
Mr Andhoey made headlines last year when he and his companion, Samuel Massie, 19, attempted to cross the South Pole on quadbikes.
The Berserk was waiting for them in McMurdo Sound when it sank.
Mr Andhoey, 35, and Mr Massie were later rescued from the ice and faced severe criticism for attempting the journey.
This week New Zealand authorities said they were looking for Mr Andhoey because it was believed he was embarking on a similar illegal trip.
Norwegian authorities had not authorised last year's voyage, or this one, as required by that country's law.
Mr Massie is also believed to be aboard the Nilaya.