The spouse working in the Republic may claim the full married persons allowance and double rate bands, but will then not pay enough tax to cover the other spouse's tax liability in Ireland.
The Northern Ireland income needs to be declared on a tax return each year, and the liability arising is based on a formula, and must be self assessed and Preliminary Tax needs to be paid as required.
By claiming only single tax credits and rate bands this should cover the liability, and in fact a tax refund may be due.
However, tax returns do need to be completed to be tax compliant in Ireland.
Pm me if you need help with your tax returns.