But the planned review didn't occur, Greenpeace said.
Russel Norman, executive director at Greenpeace, said the recording was the "smoking gun" that connected NZ corporate fishing companies to Government fishing policy, via NZ First.
"This recording from March 2018 proves what the Government has previously denied: that it was NZ First that blocked cameras on boats," he said.
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"Recreational fishers, environmentalists and all New Zealanders will be appalled by the contents of the leaked phone call in which the Minister confirms that opposition by NZ First, a party bankrolled by the fishing industry, is the reason cameras on boats were blocked."
The previous government and then head of MPI, Martyn Dunne, agreed to put cameras on boats in 2016, after the Heron Ministerial Inquiry found significant problems of illegal dumping and MPI's failure to prosecute this dumping, Norman said.
Cameras on boats were identified as essential to monitor illegal fish dumping and the illegal non-notification of the deaths of dolphins and other animals caught in fishing operations.
Norman called on the Government to remove NZ First from influence over the fisheries policy, citing the party's "history of receiving cash from big fishing companies".
"This leaked phone call makes it abundantly clear that even the Minister of Fisheries privately believes that NZ First was the key decision-maker on blocking cameras on boats."
NZ First leader Winston Peters this morning released a statement in anticipation of the story, calling it "the worst form of unethical tabloid journalism".
"New Zealand First has worked constructively with its Coalition partner on hundreds of pieces of legislation and policy, and cameras on fishing boats is no different," Peters said.
"As the Fisheries Minister Stuart Nash will attest, New Zealand First raised legitimate concerns about cameras on boats, namely their cost and who would be able to access the data.
"It is vital that sound policy has sound implementation – we are not going to be putting fishing boats, crew and families out of business because of thoughtless bureaucracy."
It was Nash's office that asked to delay the introduction of cameras on boats, rather than NZ First lobbying, Peters said.