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One agency designed them both, the Ministry of Health said, and taxpayers should not need to fund any fix.
The Southern District Health Board sought a logo for the hospital project in early 2018. The logo, with minor changes, has been used in all internal and external communications on the high-profile construction project since.
A similar logo has been used for years by The Theatre in Health Education Trust, for Sexwise, a sexual health programme for secondary school students.
The Ministry of Health, which funds the theatre trust and is a prominent partner in the hospital project, was recently made aware of the problem.
"Both the ministry and trust, understandably, have some affinity for their respective logos, but the reality is that some compromise will be needed as they are too similar for both to continue," a ministry spokesman said.
"We do not know the costs for the trust. There have been, of course, additional costs in terms of having also developed branding collateral using the logo."
The ministry chose not to name the design agency.
"We are advised by the design company that they were unaware of the trust having used the logo as they had developed a range of branding collateral for them.
"However, what is clear is that both the trust and ministry were innocent parties to this confusion, both having received logos that we thought were unique to us each."
The theatre trust, a registered charity in Dunedin, runs shows throughout New Zealand about health, wellbeing and making wise sexual choices.
Theatre artistic director Evan Hastings was confident of a logos resolution soon but otherwise declined to comment.
When the Otago Daily Times asked the ministry if referring the hospital logo back to the artist was likely part of the solution, a spokesman said the ministry was working "amicably" with parties and expected "the matter will be resolved very soon without any additional costs being publicly incurred".