Council staff said the wording had already been softened to be a "goal".
Some councillors, however, thought the wording was still too strong and prescriptive.
Leanne Brown was concerned the wording would "tie us in so tight that we eliminate a really good trustee".
"It should become a consideration not a directive," she said.
Terry Molloy said he was comfortable with the wording as it was a goal, not an absolute, and it reflected the council's desire for better relationships with tangata whenua.
Councillors rewrote the principles to read:
"While the over-riding principle is to appoint on merit, Council encourages diversity of gender, skills, thought and culture on its Council Organisation (CO) boards, and may consider the appointment of a Director with strong connections to and the support of local iwi, subject to the required skills and experience."
Tauranga City Council has three council-controlled organisations: Bay Venues Ltd, Tauranga Art Gallery and Tourism Bay of Plenty.
The revised policy also gave the council more involvement in the appointment process, with two elected members on the appointment panel.
The new policy will come into play with Tourism Bay of Plenty's recruitment of two new trustees, starting on July 1.