The whale was first seen on Monday off the coast of Kaikoura on the north-east coast of South Island.
It was not clear how it came to lose its flukes, said local Department of Conservation ranger Mike Morrissey, but "it could have been the result of entanglement" in fishing nets.
But despite what looks like a severe injury, the whale seemed to be doing fine, he told the BBC.
"It appeared to be in good health and it's bending in the water and using its pectoral fins to spin around," Mr Morrissey said.
Whales are sometimes caught in fishing gear, which can injure or kill them.
The tears appear to have happened at least a year ago, Mr Morrissey added, as the images show little sign of open wounds.
"This is a young whale and it has healed up. It doesn't seem to have affected it much at all," he said.
It is the first time anyone in the area has seen a whale like it, he added, but he held out hope it might return.
"If it's migrating, then we'll see it again," he said.
Humpback whales have begun their annual migration from Antarctic feeding grounds to breeding grounds in the South Pacific.
Anyone who spots the whale is asked to report it to the Department of Conservation on +64 800 36 24 68, so it can can be tracked.