Exeat is mainly used in British English, less commonly understood in other English-speaking regions. When I studied in the UK, I needed an exeat to leave campus.
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Formal Permission
Exeat is a formal permission, not just casual spoken consent. The headmaster provided a written exeat for her to go home.
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Temporary Leave
An exeat is granted for a specific, limited period, not for indefinite leave. His exeat allowed him to leave the monastery for two days.