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I’m in Norn Iron - we normally refer to the ROI as The South. The majority of people I know from the ROI refer to NI as The North.
Oddly your complaint about misspelling of Eire may be a result of people not being able to get the acute on the capital E.
This is from the .gov website - notably no acutes on the E’s.LIST OF PROSCRIBED GROUPS LINKED TO NORTHERN IRELAND RELATED TERRORISM
Continuity Army Council
Cumann na mBan
Fianna na hEireann
Irish National Liberation Army
Irish People's Liberation Organisation Irish Republican Army
Loyalist Volunteer Force
17 July 2020
Orange Volunteers
Red Hand Commando Red Hand Defenders
Saor Eire
Ulster Defence Association Ulster Freedom Fighters Ulster Volunteer Force
It’s because British media and governments refused to recognise the name of the country as defined by its constitution (Ireland) and instead referred to it by the incorrectly spelled Eire.
So maybe it’s nuanced to some people, but it’s an important distinction and if you ever look around right wing arenas you’ll see Eire used very deliberately.