1. if it is an Overseas Territory it isn't part of the UK as defined by the United Nations.
2. If the EU has an article clearly accepting that Gibraltar is a British territory then it matters not what.. Spain gets no say in what happens with Gibraltar... in the same way that the UK gets no say in what any Spanish territories do.
Incidently.... I find it odd that there have been many months of gnashing of teeth over the NI border.... how comes only now does anyone mention the Gibraltar border? Just another case of remainers clutching at straws to frustrate Brexit perhaps?
Not quite. before the referendum I was on the BBC trying to explain that - other than the nice and easy to solve UK/Ireland border, we face difficulties r/e the Gibraltar one.
Firstly, in the 80's I believe, Spain closed the border so anyone (including Spanish) could cross into Gib, but not return. leaving the EU means that possibility arises again, as Spain couldn't close the border under EU rules (as you allude to, the EU accepts Gib as British territory).
Secondly, a more likely outcome - especially without a deal, means that documents in addition to passports would be required to cross that border; not so much a problem for tourists maybe, but for people living and working either side of the border there looms the prospect of working and residence visas.
Again I understand there are more Spanish working in Gib than the other way round, so if they made life difficult then it's for their own people rather than Brits.
Thirdly, if the Spanish get frisky over soveriegnity, there is the outside chance of blockades (a la 80's where the only routes in were by air and sea) and an even remoter chance of an invasion. I would put the chance of the latter at next-to-zero BUT when governments in trouble (see under "Argentina" and "Falklands") are looking for a major distraction....
I would consider letting Spain share (and pay for) the airport - it's very close to the border and is possible to have two routes in and out so that Spain/Schengen flights do not have to go through the UK border; I think that would boost their local economy around Gibraltar and bring in much needed income to an expensive bit of infrastructure.