My wife is an EU national {eire}As I understand

Yes,  your wife can get her  residency and you can ask for a Permesso di Soggiorno to remain in Italy with her.   Her residency does not automatically apply to you however.    Obviously you both have to fulfil the requirements,  permanent address, health insurance, etc

if your wife is a European citizen, she can come and go in any European country as she pleases - To live in Italy, as a resident, she must apply for residence, in an Italian municipality - If you are not a European citizen, you can come to Italy for up to 90 days each year. And if you want to stay in Italy, as the husband of your wife, after he has obtained the residence, you can ask for the family reunion, but to obtain it you will have to go to an Italian consulate, of the country of which you have citizenship and ask the office. immigration, the procedure to follow.

More info to this link , of my web site >  https://www.italianmortgageservice.com/Inglese/italyresidenceguide.htm