The Irish border seems to be the biggest obstacle to establishing an agreement on Brexit and we've seen considerable talk about hard and soft borders. There are two potential solutions that I have NOT seen discussed and I just want to float them here to see if there is any chance that either of them might present a possible solution. I expect both would be seen as somewhat "radical" in the sense of being a rather big step from the status quo but my knowledge of Ireland is far too meager for the reasons to be obvious to me.
Solution 1: Have Northern Ireland merge with the Irish Republic so that the entire island is the Irish Republic. In that scenario, ALL of Ireland would be in the EU and the Irish border would disappear. Great Britain, comprising England, Scotland and Wales, would be free to leave the EU and Ireland could stay in the EU.
Solution 2: Have all of Ireland join the UK on the same basis as England, Scotland and Wales. All of the enlarged UK could then leave the EU together. (Mind you, I could imagine this giving a lot of ammunition to the people who want a second referendum and if that happened, perhaps Brexit would be abandoned and the enlarged UK would remain in the EU after all. After all, the people in the Irish Republic could rightfully say that they weren't even consulted about Brexit.)
I'd be very curious to know whether a reunification of Ireland - either as a single Republic or as part of the UK - is even imaginable or are the differences still so strong that it's unthinkable? From what I hear, the inter-Irish border is all but invisible since the Troubles ended and the two parts of the island get along fine. I also sense that the Catholic and Protestant Churches are far less influential than they were; after all, the Republic even permits abortion now, something that was unthinkable just a few years ago. The violence has either stopped entirely or declined a very great deal. The current prime minister in the Republic is apparently gay and only half-Irish, another sign that attitudes in the Republic have greatly changed.
Could Protestants in the North be comfortable in an enlarged Republic? Would they need certain safeguards to even consider such a move? I would certainly not want to make any such move with the consent of a majority of those in Northern Ireland.