The breakaway FAI was not a response to partition. The Anglo-Irish Treaty, agreeing partition, was signed in November 1921. The FAI was formed in June 1921. While initially seeing itself as a rival to the IFA - a sort of American League to the National League - with hindsight, it was more an attempt to divide football along national/sectarian lines - 1923 FAI Cup winners Alton United were from Belfast. While shades of this remains, the FAI was forced to fit itself in with partition by the Liverpool Conference, and rename itself the FAIFS.
Incidentally, it's not just a shared history, but shared shirts too. Both Irelands before and immediately after WW2 wore identical 'rugby' shirts in games with the shield and shamrocks as the badge.
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