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Hard-Brexiteers would like the UK to have a "clean break" with the EU, and in particular they don't want any part of the UK to stay even temporarily in the customs union (the so-called backstop).

This "clean break" would normally require a hard border between Northern Ireland and the Republic of Ireland, but a hard border would breach the Good Friday Agreement. It seems that the options "the UK leaves the customs union" and "the Good Friday Agreement is upheld" are mutually exclusive.

As far as I'm aware, the only idea that some hard-Brexiteers have proposed to solve this contradiction is a soft border using "seamless technology". Even assuming that such a technological option is feasible for the trade of goods, wouldn't that leave a backdoor wide open for illegal immigration from the EU to the UK? If yes, are hard-Brexiteers ok with that? Alternatively, are there any hard-Brexiteers who support a hard border?

To summarize, what do hard-Brexiteers see as the ideal outcome for the Irish border in the long term?

They tend to think it's somebody else's problem (Ireland's and/or the DUP's). Unless you are part of the DUP of course. See how Rees-Mogg has been punting the problem along the lines of: I agree with whatever the DUP agrees (or at least doesn't oppose) on Northern Ireland. And at the same time he says that in the case of no-deal Ireland would not dare to impose a hard border. Which is true to some extent.

The DUP has given a number of somewhat contradictory statements on this, over time. From the somewhat famous denial that a hard border ever existed to the more recent position(s) that they would prioritize staying the EU over splitting Northern Ireland from the UK.

I'm a Remainer, but have been accused of being a Hard Brexiter on some stances in this debate, so I'll take a stab. The WTO, the Repubic of Ireland, and the EU have all recently made declarations and supporting statements that they have no intention of putting up border checks on any border between the Northern Ireland and the Republic of Ireland. They will instead impose checks at warehouses and other centers of of commerce. The UK is adamant that they will not impose any border check on any borders between ROI and NI (already been there and done that) and currently, through the HMRC, conduct such checks (for immigration and contraband) in Northern Ireland. So it seems that this issue is, perhap ironically, sorting itself out whether there is a 'hard' Brexit or not. The means by which such controls can be done are long established by many nations around the world (a simple Google search with near infinite examples, though if enough people insist, I'll post all sorts of links to customs seizures and immigration raids away from any borders). It will not be perfect. Nobody has a perfect system that I'm aware of. But it will suffice to control immigration and movement of goods to some extent and, most importantly, prevent any return of the Troubles.

The brexiteers don't really want anything regarding the Irish border. It's just a problem preventing them getting the hard brexit that they want, and since they don't have a real solution for it they just want to pretend it's not really a problem.

That's all it is, an annoying roadblock for them.