My brother in law is in a similar situation as the OP. He's italian national and despite he's married with a Irish citizen (my sister) he's not Irish domiciled. He seeked professional advice on the remittance basis and I was with him when we met the tax advisors. The meeting covered a lot of aspects but this is a brief summary
1) It's quite difficult to obtain an irish domicile. He would have to severe all ties with his original country, sell the apartment that he still owns, close the bank account in Italy and prove that he has no ties anymore with Italy, he declares that he never wants to return on a permanent basis to Italy, otherwise he would still be deemed italian domiciled even if he had a Irish passport (he only has an Italian passport by the way).
2) Needs to avoid transferring foreign assets to his wife's name (since she's Irish domiciled and Irish tax resident) and some transfers would be seen as a tax avoidance move so that the wife could benefit. This doesn't apply to the OP since the wife is not Irish domiciled
3) Don't remit any foreign money into Ireland not only by bank transfer via bank but also avoid using the credit card linked to his Italian account in Ireland while he's tax resident in Ireland for large purchases.
4) Not all investment vehicles can avail of the remittance basis, but since he only has shares and U.S. domiciled ETF with a U.S. broker (he lived in the U.S. before moving to Ireland) they do fall into the remittance basis scheme
5) What Fergal19 refers to:
"Are you looking to acquire these assets from funds accumulated while you were in Ireland? If so Im not sure if this would work due to the transfer of assets abroad rules. I think there is anti avoidance for Non Doms which will deem the income as Case IV and therefore not subject to the remmitannce basis of taxation"
was also clarified for my brother in law and only applies to specific situations: for example if he holds 100 shares of Kerry Gold with AIB, he sells the shares and makes 10.000€ capital gain , pays the tax due in Ireland on the 10.000€ then he can do whatever he wants with the rest of the money gained as it's clean money, taxes have been paid: if he transfers the money abroad and reinvests it and makes further profits from that money then it doesn't matter, he can avail of the remittance basis all the time.
If he owns 100 shares of Kerry Gold with AIB, transfers those shares into an account in Dubai today, sells the shares tomorrow and makes 10.000€ to avoid paying the irish capital gain tax then this is not seen as "foreign income" it may be seen as a move to avoid paying taxes in Ireland. This is what Fergal19 refers to.
Seamus