"mutual understanding of" is correct and usable in written English.
You can use it when two parties need to come to an agreement, or when two or more parties need to have a shared understanding of something. For example: "We need to find a mutual understanding of the challenges facing us."
The center brings together scholars, judges, officials, and lawyers to deepen mutual understanding of both societies.
The organisations came to a mutual understanding of the scope of the programme, its deliverables and outputs.
Countries don't change their behavior through mutual finger pointing, but through a mutual understanding of respective failures.
Their romance blossomed quickly, as they shared a love of traveling and a mutual understanding of each other's unusual schedules.
Shared mental models are crucial for constructing mutual understanding of the patient's condition during a clinical handover.
The goal of the research is to develop a mutual understanding of common values and so support future developments.
If we can build some mutual understanding of how each sector works and what they are trying to achieve I think a lot of progress could be made.
The issue was to make sure there was a complete understanding of our support for the engagement policy and the mutual understanding of the nature" of North Korea.
You've got to have a mutual understanding of what everybody's objectives are because life ain't a bed of roses and we should debate and discuss".
A result, he added, was "a lack of mutual understanding of what the Reynolds project would be before the Smithsonian agreed to the gift".
It's time we learned to speak up and address these interruptions and safely build mutual understanding of their appropriate use in our life.
