It is an open question whether the Irish Republic can remain in the EU, when the UK is not in it.
In the 1970s, the Republic found it necessary to follow the UK into the EEC, on January 1st, 1973. The UK is the Republic's most important trading partner, and the Republic and the UK had a common currency until 1979.
Now the Irish border is a concern, and the only possible solution is not to have one. But this means the Irish government may have to bow to the necessities of history and follow the UK out. One out, all out.
The Irish Free State must once again become free.
The Republic and the UK are both committed to the Good Friday Agreement, whereby there is no 'hard' border, i.e. no physical checkpoints.
One method of achieving this is for the EU to be removed from the equation, so that the UK and the Republic can resolve the matter bi-laterally. This requires the Dublin government to make a bold move, and leave the EU.
There is no disagreement between London and Dublin; the Good Friday Agreement has been long in effect. The EU is now part of the problem, and so the next move is up to Dublin. But so far they have not yet recognised the changed political and economic situation they find themselves in.
The EU is unhappy about this possibility: all of its manoeuvering is designed to edge the situation away from this, mainly by seeking to control the UK's actions in perpetuity, by locking the UK into an agreement with no exit provisions, in which true control of the border will be handled (permanently) by Brussels.
The security situation in Ireland can only work if the Republic and Northern Ireland are both on the same side, and that includes the same side of the Brexit arrangements. It only works if both states are in the Eu, or both are out of the EU. It won't work with interference from Brussels: there is no squaring the circle if the EU is free to impose its arbitrary rules, full of conditions that simply can't be met.
An end to EU interference in the problem is a sine qua non of future progress.