Officially, the name of the state is Ireland as per the constitution, Republic of Ireland is a description introduced in the Republic of Ireland Act, (1948). But you're right in an English conversation, it's not Eire.No it isn't, not when the rest of your post is in English it's not. It is the Republic of Ireland.
Not sure what your angle is using it in this context, but as it is nothing to do with the Re-turn scheme, I'll leave it there.