Hi Hdsd,
Non UK resident employers do not need to register as employers in the UK.
This means that they do not need to register you as an employee in the UK.
They need to do this in Ireland.
You will need to request a certificate of residence from:
How to apply for a certificate of residence to claim tax relief abroad
So that you can claim with the Irish tax authorities to have your employment income paid tax free, as you are liable to tax on this in the UK.
You complete a Self Assessment tax return to declare the foreign empoyment income.
Thank you.