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However, I would recommend rephrasing the sentence as "have a shared desire" or "share the same desire" as it sounds more natural and concise.
Yes, the phrase "share a common desire" is grammatically correct and can be used in written English. It means that multiple people or groups have a similar or shared desire for something. Example: The protesters and the government officials may have different opinions on how to address the issue, but they all share a common desire for peace and progress.
Exact(5)
Recreational and elite athletes share a common desire to improve their skills and to win, rather than lose, a contest.
Although most doctors stay outside the political debate, they share a common desire for a safe humanitarian corridor and to be able to carry out their medical work without a fear of being targeted or their hospitals being hit.
Islamic movements in the five countries between Casablanca and Cairo share a common desire to get back to the roots of their faith, a destination of the governments they live under, and a hot dislike of what they think of as western policy.
They are the world's largest stateless minority, and although they are divided by nationality, dialect, custom, allegiances and religion, they share a common desire: to be able to express their ethnic identity and to govern themselves in the areas in which they live.
US President Barack Obama and Russian President Vladimir Putin acknowledged that, though they share a common desire to end the violence in Syria, they have a different stance on how to approach the conflict.
Similar(53)
In spite of their differences, they shared a common desire: to speak and be heard.Political wrangling, however, had already marred the meeting before it even started.
The leader of the Roman Catholic Church said the two men shared a common desire for social justice, peace and the promotion of Christian values in areas such as marriage.
Pope Francis said they shared a common desire for economic and social justice.
Both men acknowledged their differences but said they shared a common desire to stop the bloodshed.
All shared a common desire to promote 1st‐year retention for new graduates.
Although each study shared a common desire for measurement of a particular digital skill, the objective of each study was to answer a specific research question.
Related(20)
share a common wish
share a common passion
share a common commitment
share a common intent
share a common interest
shared a common desire
share a common vision
share a common future
share a common selfishness
share a common assumption
share a common surname
share a similar desire
share a human desire
share a common challenge
share a legitimate desire
share a fierce desire
share a common bond
share a common characteristic
share a political desire
share a common language
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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