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The phrase "agree mutually" is correct and usable in written English.
It can be used when indicating that two or more parties have come to a consensus or shared understanding on a particular matter.
Example: "After much discussion, we were able to agree mutually on the terms of the contract."
Alternatives: "reach a mutual agreement" or "come to a consensus".
Exact(2)
Of 22 cases in which the BBC made extra discretionary payments totalling £510,000 the NAO said 16 were made as part of a "compromise agreement... to avoid possible disputes and agree mutually acceptable terms".
The distributions of surface wetness obtained from both methods agree mutually with each other.
Similar(56)
His subsequent resignation was agreed mutually between Mr Longworth and the BCC board and there were no external factors involved.
Allison's departure from Ferrari was agreed mutually but his position had been the centre of speculation for a number of weeks.
Hillary Rodham Clinton and Representative Rick Lazio both claim credit for taking the initiative and agreeing mutually to ban soft-money broadcast ads in their New York Senate campaigns.
Osama al-Ali, a senior Palestinian security official, said the Israelis and Palestinians had agreed mutually to "exert maximum effort to stop the fire".
"Our interest is that the Global Fund and the Cambodian government agree a mutually acceptable way forward," he said.
"When we're busy, we agree to mutually ignore each other.
In June last year, nine of the biggest TV manufacturers declared a truce and agreed on mutually acceptable prices.
We've agreed a mutually beneficial deal with Shrewsbury".
The Robins agreed to mutually terminate his deal so he could sign a two-year contract with the Bantams.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com