The Greens have traditionally opposed waka-jumping legislation, voting against previous versions of the bill in 2001 and 2005.
The current bill, now before select committee, would preserve parliamentary proportion as determined by the voters and would mean that an MP would have to leave Parliament if the MP left a party or was expelled.
The bill is part of the Labour-NZ First coalition agreement, but with National opposing it, it won't pass into law without the support of the Greens.
Genter said there had been concerns among the membership about the party's support for the bill at the first reading.
"I don't think we've done as good a job as we could at communicating why we're doing what we're doing, or demonstrating that we are listening to the membership.
"Around our deliberations and position on the waka-jumping bill, I think we could have done a much better job at engaging with members who were raising concerns. We should have engaged earlier in a discussion about whether or not we should support it."
She said she was happy to vote for the bill at first reading, but the party needed to have a "big debate about what we do at second reading".
Former Green MP Sue Bradford, who is no longer a party member, applauded Genter's courage for publicly speaking out on the bill, which Bradford strongly opposes.
"There are quite senior people in the party who are really unhappy about the party's total reversal on its position. Worst of all, it's a rejection of the Green party's own history," Bradford said.
"She's really making a case to differentiate from the Government, that the Greens don't have to be a lapdog or a doormat.
"It could be fatal for the Greens if they don't find themselves able to challenge Labour and NZ First in general, but this is just an extraordinary example of them caving in without listening to older party members."
Former MP Keith Locke, who is still a party member, has also criticised the party over its support for the bill.
He did not have a good gauge on the depth of feeling of discontent among the membership, but said it was an important issue and looked forward to debating the issue with other party members.
Marama Davidson has also made a point of carving out an independent voice, but used the TPP as an example - a less controversial choice as the caucus and party membership are more unified on the issue.
She has said being an MP and not a minister made her better suited to speaking out against the Government, but Genter said publicly disagreeing with a Government position as a minister could be even more powerful.
Davidson said she was not available to be interviewed today.
A full list of candidates for the co-leadership will be announced on Monday. Delegates will vote in late March and early April before a winner is announced on April 8.
