Northern Ireland is a country that is part of the United Kingdom. The Republic of Ireland is its own country that is not connected to the U.K. politically. What do the people of the Republic of Ireland call the northern part of their country (Sligo, Dundalk, etc.)? “Northern Republic of Ireland,” maybe?
Mostly “Northern Ireland”, same as the inhabitants thereof. Sometimes just “the North”. Occasionally “Ulster”, though that’s not actually accurate: Two of the eight counties of Ulster are still part of the Republic of Ireland.
Why would they make up some new name?
My experience is that whatever is being discussed is often in terms of the nearest major city or the County. “In Down” or “In Newry” refers to anything happening near my family home.
And of course people in the ROI call the city of Londonderry “Derry”.
Usually “The North”. More formally “Northern Ireland”, or if we’re trying to be funny “Norn Iron”.
But nine times out of ten, in normal conversation it’s “The North”.
Edited to add: I misunderstood, you’re asking about the northest parts of the Republic of Ireland. Sligo and Donegal are sometimes referred to as the Northwest. But mostly Dundalk, Drogheda, Cavan, Donegal etc. are simply referred to by their individual town or county names.
I always wondered how it happened that part of Ulster split off to become part of the Republic of Ireland.
Now that I think of it, the areas in the northern part of the Republic are often referred to as the Border Counties.
IdidnotreadtheOPistan?
“Northern Ireland as defined by the Act, amounting to six of the nine counties of Ulster, was seen as the maximum area within which Unionists could be expected to have a safe majority. This was in spite of the fact that counties Fermanagh and Tyrone had Catholic Nationalist majorities.”
That section probably needs a cite, but for the most part the inhabitants looked towards the UK rather than the proto-Republican agitation.
I read the OP, but misunderstood it.
Minor correction: It’s three of the nine counties of Ulster: Cavan, Monaghan and Donegal.
You got me there. I remembered 26+6, but assumed incorrectly that the Four Provinces each had the same number of counties.
So then how did Fermanagh and Tyrone get stuck with N.I. anyway?
Anything much smaller, it was feared, would be too small to be economically viable as a state.
Because it was believed a four-county Northern Ireland wouldn’t be viable as a state, the other four counties were majority Protestant.
Although even those had areas with a Catholic majority, I grew up in one in County Antrim.
Just concerning nomenclature: I understand that people in the Irish Republic often refer to Northern Ireland as “The Six Counties” – a fairly neutral and un-inflammatory “shorthand” expression; also I believe, sometimes used by Northern Irish folk for the same purpose.
That would be the city administered by Derry City Council, served by City of Derry Airport and home to Derry City football club and City of Derry rugby club?
Its usually ‘The Occupied Six Counties’ in my experience 
As Galwegian says, the areas that border Northern Ireland is mostly called “the border counties”. It’s not a phase in very common use; the areas concerned have little in common except the fact that they border NI. Does the US have a term for the states that border Canada?
Basically, they tried to construct the largest area that would still have an overall Unionist majority large enough to be securely Unionist. The result was a Northern Ireland that contained substantial areas with a Nationalist majority, while virtually no signficant part of the Irish Free State (and certainly no entire county) had a Unionist majority.
The people in the city mostly call it “Derry” too, unless they wish to Make A Point.
So did my great-great grandparents. By another line I’m also descended from McGuires of Fermanagh.
Yes. The Border States or Upper Midwest, depending on which part.