What's the adjective for the Republic of Ireland (Eire)?

Solution 1:

The name of the island and its parts is subject to complex cultural and political questions, as you've surmised. Even the British Media cannot agree on what names to use.

If you're looking to identify distinct regional accents on the island of Ireland you might be better off referring to the counties, or at least to the historical provinces: Leinster, Ulster, Munster, Connacht. Those correspond to the broad areas of regional English accents.

If you do need an adjective to refer to the Republic of Ireland, avoid Southern Ireland (capitalized) since that was briefly the name of a UK political constituent and could be confusing. You should similarly avoid using the Gaelic name Eire in English. One possibility is to refer to the Republic of Ireland in contrast to Northern Ireland, as I have done here. Some people still won't like even that, I suppose.

If you insist on having an adjective instead of using "of"-consructions, your choices are limited even further. Consider referring to the counties/provinces if you can, it's more exact and much less subject to dispute.

Solution 2:

The adjectival form of Ireland is Irish. For this and other geographical names, see this document, published by the UK’s Foreign and Commonwealth Office.