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Firstly the model you are using appear not to tax efficient - ie using a UK company to subcontract work in Ireland as you will likely be taxed in both countries subject to double tax credits.
What work are you doing and how many days are you spending in UK and Ireland?
Where are you resident/ whats your tax domicile?
I work Monday To Friday. I fly to IReland on monday morning and return on Friday evening. I spent all weekends in UK and all annual holidays and bank holidays in UK. I have my family and home in UK. In Ireland I reside in B&B
I am UK resident
In that case you are tax resident in (1) UK as your family/and home is in UK and you spend a subtantial time in UK (ii) Ireland as you also work and spend time in Ireland.
In that case you are liable for UK and Irish tax. Depending on how your contracts in Ireland work, you should register for tax in Ireland and pay tax on the income earned in Ireland. When completing UK tax returns, the tax already paid in Ireland can be allowed as a tax credit.
However if the the contract is for 6 months or less and you are Irish non-dom, then you may not need to pay Irish income tax.
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Other issues to worry about are VAT if you are VAT registered. As the supply of services is to a business customer, then the "place of supply" is where the business customer belongs, i.e. Ireland and outside UK VAT scope and no VAT is charged. On conclusion there a number of ways of making your structure more tax efficient and I shall list them below:
Option 1: If staying less than 183 days in Irish tax year. It best to keep the way things and use your UK firm providing services to Ireland. You are unlikely to pay tax in Ireland.
Option 2: If you are staying longer, you may need consider using a smaller agency or umbrella companies or one ideally in offshore centres. OR
Option 3: If you are staying longer, you can set up your own Irish LTD. There is lower tax in Ireland. However there are things to watch like ensuring you comply with HMRC offshore regulations and anti-avoidance tax laws. In addition Irish companies have other specific requirements like >1 director etc.
Option 4: If you are staying longer or permanently working there, then there is no point in having a UK company rather maintain an Irish Company instead.
Option 5. You can use offshore schemes. Most people who work across jurisdictions find setting up a company in places like Channel Islands or low tax jurisdiction makes sense. The very fact that you are providing international services justifies that- however you also need to comply with all the countries you provide the service to ensure you are not outside their scope.
My final comment to consider the above, but before you make any decision on the options please hire an offshore tax adviser whom you will discuss in detail of your intentions and he will advise on the risk and rewards of each option above.
Hope this helps.