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The phrase 'a mutual interest' is correct and usable in written English.
You can use it when referring to a shared interest between two or more people. For example, 'My colleague and I had a mutual interest in reading literature.'.
Exact(59)
Thus, the two purported gladiators were in business together, with a mutual interest in making the interviews interesting enough to make a nice profit.
Combating terrorism is a mutual interest.
After his release, there was a mutual interest with Hofstra.
"We have a mutual interest," says the scientist.
Many analysts and businessmen say there is a mutual interest in having Iskandar flourish.
Kaine was there with his harmonica, and the two discovered a mutual interest.
They share a mutual interest in reaching an agreement rather than letting Reed hit free agency.
At the time, the Rocherolles were a young married couple with a mutual interest in horticulture.
So diversions are always welcome, even diversions driven by a mutual interest in positive publicity.
They affirm a mutual interest in providing security, and recognize differences on certain issues.
Similar(1)
A teacher in California noticed one of her students who wrote "environmental protection" on her sticker--a mutual interest for the two.
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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