Equip the butterfly emblem once you get it as well as avoiding Side-Op 51 if I remember right. Assuming Quiet is one of your companions.
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The butterfly emblem is not needed. It stops your progression, and you can just get her back by doing her mission a few times again.
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Fun fact: after you get her back, she takes up a spot in your max MB population, and since she's always in the brig, it means one of your MB units will be unable to have the max number of people.