An Irish passport is issued by the Republic of Ireland, not "southern Ireland". The Ireland Act of 1949 ensured that any Irish citizen might become a citizen of the UK simply by moving from the Republic to the UK. Since the passage of that act in 1949, and especially since the Irish Nationality and Citizenship Act was passed in 1956, the Irish Republic has been granting Irish passports to residents of Northern Ireland who apply for them.

Nonetheless, I have never seen the term "British-Irish" used. One frequently sees the term "Anglo Irish" used to describe Irish citizens whose ancestors are English, the same way one sees the terms "Anglo American" and "Irish American" used for US citizens with English and Irish ancestries, respectively.