I'm still confused how this is going to work for Northern Ireland.
It's lovely that she values the historical and familial ties between Britain and Ireland, but I still fail to see how the hell we are going to keep the CTA (Common Travel Area) working. Effectively, all that ever guaranteed was freedom of movement of people and labour.
Ireland and the UK recognise eachother's citizens as non-aliens. So, basically if you're Irish in the UK or British in Ireland, for all intents and purposes (including voting rights in parliamentary elections etc) you're treated as a local, in a way that goes far beyond EU rights.
They're not even required to produce ID - although, for practical security reasons in airports it happens.
However, on the land border between the two countries it's not even marked. It's as visible as the border between two counties in England and probably less marked than the England-Scotland border as we tend not to play it up. The only way you would realise you'd crossed is the road markings change from a white line on the edge of the road and (ironically) standard European type signs in UK (Northern Ireland) to an orange line at the outside of the road, US/Aussi/NZ style orange warning signs and km/h in the Republic and your mobile phone might alert you to roaming.
HOWEVER, that never applied to to the movement of goods, services or capital and that's where this lovely sentiment all falls apart. If the UK is not part of the single market or the customs union, we end up with a customs border which will be exactly the same, if not worse (as if the UK doesn't join the customs union) as the EU-Turkish border.
While it means that your average Irish or British citizen can pass freely across the border, if you have a bag of shopping with you from the local supermarket you could be done for smuggling.
Also since the 1990s, Irish and Northern Irish economies and infrastructure have begun to become shared and deeply integrated. For example the electricity networks are owned and operated by the same companies, interlinked at various points at just regular transmission voltages and share common standards. The telecoms networks are largely intergrated in a very local way and have been for many decades - a local call is a local call. The Republic had agreed to fund road infrascture in the North where it facilitated interlinking regions like Donegal (extreme Northwest of Ireland in the Republic). There are loads of examples of this kind of thing.
Many, many businesses also just operate as if there's no border at all. So, everything from logistics to supermarkets, to agrifood, to you name it tend to operate on an all-island basis. Cutting Northern Ireland off from the Republic at this stage could be hugely damaging to the North probably more than the Republic as it's the smaller and less financially flush partner in the relationship. A lot of Northern SMEs are very dependent on having access to the rest of Ireland for trade. So, I would assume the UK is planning a sizable buffer fund while it readjusts its economy, to avoid it spinning into a depression?
There had also been growing integration of things like health services and you've various cross-border bodies that are effectively QANGOs for areas like tourism, food safety and so on.
You've also got a North-South Ministeral Body and the British-Irish Parliamentary Assembly, which is pretty unique in any EU context.
On top of that you've now got BVIS - British Irish Visa Scheme that allows mutual recognition of certain classes of business traveler visa so that citizens of China and India can travel to both countries on a single visa. It's a little like a mini-Schengen between the UK and Ireland.
How the hell is all that going to work after Brexit?! An awful lot of it is predicated on both countries being EU members.