I don’t think people who aren’t from this island realize the extent to which Ireland really is one nation, even if it’s two different jurisdictions. The vast majority of sporting, religious, and other cultural bodies in Ireland operate on an all-Ireland basis.
In sports like rugby, there is a single national team for the whole of Ireland. In sports which don’t generally have national teams, such as golf and showjumping, the organization is nonetheless on an all-Ireland cross-border basis. Association football (soccer) is an exception. (I think the sole exception, though I’m not sure about sports like gymnastics.) The Gaelic Athletic Association also of course operates on an all-Ireland basis, and that too doesn’t generally have national teams. (There actually are national teams, in International Rules Football, Compromise Rules Shinty-Hurling, and Rounders, but none of these contests are the pinnacles of their sports.)
The Irish Shows Association, which organizes agricultural shows, is all-Ireland. So is the Association of Irish Musical Societies, the Railway Preservation Society of Ireland, and many others. On the Wikipedia category list “All-Ireland organizations“, I also see the Irish Cave Rescue Organisation, the Irish Federation of Astronomical Societies, the Irish Museums Association, the Geographical Society of Ireland, and the Irish Council of Churches. There are also a few trade union bodies: the Irish Congress of Trade Unions, the Association of Secondary Teachers of Ireland, and SIPTU, and one professional body: Accounting Technicians Ireland. There’s also the Royal National Lifeboat Institution, which also operates in Great Britain, the Channel Islands and the Isle of Man.
There are even government departments which operate across the island, under the aegis of the North/South Ministerial Council. One that I am aware of is Waterways Ireland, responsible for the management and upkeep of internal navigable waterways (canals and rivers) across the island, because their headquarters on the Grand Canal are in Tullamore, a stone’s throw from my house. There is therefore a sign up on Harbour Street in three languages: Waterways Ireland, Uisceabhealaí Éireann, Watterweys Airlann.