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The word "three-way" is correct and usable in written English.
You can use it to refer to something that involves three parties, such as a three-way agreement between two countries and a third mediator, or a three-way conversation between three people. For example: "The negotiations resulted in a three-way agreement between Russia, Ukraine, and the European Union."
Dictionary
three-way
adjective
Relating to three different directions.
synonyms
Exact(55)
Now here's Jim's three-way dilemma.
It's a voluntary three-way deal.
She stitched together a three-way coalition.
Obama's Disapproval Ratings: A Three-Way Street?
"It was a three-way package deal.
This valve may be two-way or three-way.
Similar(5)
The two-way street.
So, two-way interaction.
Friendship two-way street.
Two-way hardware multithreaded here.
Relationships are two-way streets.
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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