How about the USA? The USA has a large and influential Irish-American population and it is not impossible to imagine some alternate timelines in which peacefully or otherwise, US influence in Ireland increases to the point that Eire is annexed and admitted to the US as a state. Technically, this would not quite be the same as the USA "ruling" Eire, but rather Ireland becoming part of the USA.
Careful! that's exactly the scenario I joined AH to promote!
The contempt was withering.
It was very much a long shot; the thread was a challenge to come up with a US colony in Europe before 1914, presumably so the USA would be drawn into WWI from the get-go. People seemed to be liking Heligoland when I last looked into that thread!
But yeah, the US had a special interest in Ireland around 1900. Not an unambiguous one, and it's just about impossible to imagine a scenario where the British would prefer it come under Yankee protection (I suggested, as a Commonwealth like Puerto Rico is today); if the revolutionary situation on the island got so bad they had to admit they couldn't hold the place they'd probably much prefer the current situation, with it a fully independent country (kingdom, republic, whatever, from the British point of view).
I had it be a TR negotiated thing; US freedom of religion finessing the whole confessional brawl problem and keeping the island unified.
Regarding a Norse or Angevin/Plantagenet regime--well, I don't see it as a direct address of the OP if Ireland winds up under the same rule as England does, even if that's a capital based in a third country. A partial at best. That's why I downplayed the cases where perhaps Ireland is indeed used as a base to subdue England from. A Nordic Britain that also rules Ireland will still be Ireland under the same rule as Britain; a Nordic kingdom that holds Ireland but fails to attempt to rule Britain would be a different matter.
So that's two paths toward the OP challenge, and my favored one is also a partial, as it depends on Ireland having standing as a unified realm that can defend itself so it's not under any foreign flag, just allied to them. Vice versa for Ireland to be independent of Britain any time before the 20th century they'd need strong allies to hold off the British with.
So the alternative is, someone acquires Ireland as a base to attack Britain from, and then fails to succeed in that attack but manages to avoid losing Ireland in the retaliation.
The tough thing about that is, Britain is right there, more or less between Ireland and any other European power. A Nordic realm might have an edge if they have Scotland in their quiver or at their side too, then the English can't try to block a sea route in. (Assuming both sides have navies, or at least the Nordics do).
And so, with Ireland at point-blank range like that, it's a pretty delicate balance of power to maintain for many generations, to have England strong enough to resist a great-power backed invasion from Ireland, yet not strong enough to take Ireland from that third power.
The Irish being fanatically loyal to the not-England side would help stabilize the situation.
But that leads us again toward an independent Ireland that is merely allied to some strong third power, not incorporated in it.
I had yet another wacky scenario in response to another wacky challenge--someone wanted a Muslim, but English-speaking, North America. I came up with some scheme whereby a strong Iberian Islamic state conquers Britain via an alliance with Ireland--the latter is very much a client state of the Muslim power, but nominally independent (and Celtic Christian). The Irish are as it were front men for the Iberian power, who industrialize Britain as a colony and indenture/enslave lots of English people and ship them overseas to the American colonies, where they Islamify as a stratagem for getting along and getting ahead, but remain English-speakers.
Wacky challenges, wacky answers.
The basic economic weakness of Ireland is probably a very good explanation for why it has been such poor soil for medieval nation-building. Still think it's odd a bunch of Normans couldn't pull it off though. They'd do that with nominal subservience to the English crown, but if they managed it I can easily see them breaking away on some pretext or other, probably dynastic.