https://wiki.openstreetmap.org/wiki/Tag:access%3Dpermit
https://wiki.openstreetmap.org/wiki/Key:access#Road_with_restricted_access

While bases are technically permit only access, lots of private communities also are, but aren't classified as such. As access into bases is difficult, classifying them as private is much more fitting than permit. The tag's used for where permit is more of a formality, rather than a secure and patrolled military base. 