Well it doesn't exist as a state in the first place, so it will be unrecognised. Some people in Ireland and Northern Ireland want it to exist - a United Ireland in effect, but it doesn't and never will, in my opinion because 'the other side' are loyal to the Queen and don't want that.
The boring history (
):
The island of Ireland has 32 counties: All 32 were united before Britain colonised the country in medieval times. In 1919, Ireland won its independence finally from Britain. 6 of the 32 counties - all in the provence of Ulster wanted to stay under British rule i.e. loyal to the queen. So in the Treaty of 1922, it was agreed to give the other 26 counties freedom to rule themselves and the other 6 would be ruled from London (giving rise to 'Northern Ireland'). Some people would argue it was based on religious grounds where those 6 counties were about 66% protestant. For the next 50 years, nothing changed except *apparently* religious prejudice where Protestants were favoured. The Catholic community had enough and wanted a United Ireland(they had always wanted this). The IRA started a battle to get this United Ireland. The people who backed the IRA, backed the idea of a United Ireland.
Now the Provisional IRA has ceased to exist as a paramilitary wing and Northern Ireland is now trying to rule itself whilst still being UK soil - like Scotland and Wales - as a devolved Government. In order for this to work, diplomats from Dublin and London had to work together to bring the two sides together
The Irish Republic is a term used by some people to indicate their opinion that all 32 counties should be united and that if they were, that is what the island would be known as. They are often in support of the IRA, but not necessarily. They often refuse to recognise 'Northern Ireland' which does exist.
Basically, the split:
The Republic of Ireland = 26 counties on the island, independent state that exists.
Northern Ireland = 6 counties on the island, however UK soil, attempting an albeit devolved Government, that exists
The Irish Republic = A term given by some people to describe the island if all 32 counties were united again, but that doesn't actually exist.