intersociety

adjective

in·​ter·​so·​ci·​e·​ty ˌin-tər-sə-ˈsī-ə-tē 

variants or inter-society

: occurring between or involving two or more societies

an intersociety council

… was chosen to represent the society against the Webster Literary Society … in the annual intersociety debate.

—Dennis K. Boman

Word History

First Known Use

1839, in the meaning defined above