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James Cockle, a software developer from Dunedin, is set to challenge Shaw for his position as male co-leader when the party re-elects its leaders at its AGM in August.
Cockle, who is campaigning largely on existing Green Party policy, is unhappy with the progress the Green Party have made this Parliamentary term.
The public knows the country is "heading up against environmental limits" Cockle said.
"People understand this, and they want real leadership and they want direction. And I think it's time that Greens offered it to them."
Cockle said it was time for the Green Party to become a "major party" and to stop being "Labour's little helper".
He said many would say this was a sign the Green Party was in disarray or a mess.
"To that I say this: the tree does not know how tall it is, you know, it just knows ... where the sun is. It knows where the light is, and it grows towards the light."
Shaw has been the party's male co-leader since 2015 and has been Minister of Climate Change and Associate Minister for Environment and Biodiversity since Green Party members agreed to a co-operation agreement with Labour after the election.
The party's other co-leader Marama Davidson was handed the Prevention of Family and Sexual Violence and Associate Minister for Housing (homelessness) portfolios.
Shaw said he was "quietly confident" he would stay in the job and rejected the idea a growing number of Green Party members did not support his leadership.
"When you look at the reasons why he's standing, they are some of the concerns that people have always had about the tensions of being in government, those tensions have been there since we went into government, but every time we've gone into government we've had the full backing of our party."
Davidson said she absolutely supported Shaw.
"I've had the inside privilege of seeing what James is working on, seeing how hard he works every single day and the concerns that are being raised here are the very concerns we have. We're always very clear, we need to push further and faster to achieve our goals."
Green Party members told RNZ the challenge was surprising but they were interested in what having a co-leader outside Parliament could mean for the party.
Some also feared the challenge could widen the divide between the left and right within the Greens.