score:2
The UK and the Republic of Ireland (commonly just called Ireland) share a land border with each other on the island of Ireland, with the Republic of Ireland to the south and the Northern Ireland (which is part of the UK) to the north. It is possible to cross that border without any immigration checks. They have a close relationship and have a free-movement agreement called the Common Travel Area allowing UK and Irish citizens to move and work freely between the two countries - Common Travel Area guidance - GOV.UK
Although those rights do not apply to non-citizens, the Common Travel Area does include co-operation with regard to immigration which applies to non-citizens:
The Common Travel Area also involves some co-operation on matters relating to immigration issues. A non-EEA national, for example, may be refused permission to enter Ireland if they intend to travel onwards to the UK and they would not qualify for admission to the UK. Irish immigration officers have the power to carry out checks on people arriving in the State from the UK and to refuse them entry to the State on the same grounds as apply to people arriving from outside the Common Travel Area. These checks are carried out selectively. (Reference: Common Travel Area between Ireland and the United Kingdom - citizensinformation.ie)
As such, due to this special agreement between the two nations, you should expect closer inspection when entering either of the two if you have admission refusals or restrictions on travel to the other. They may suspect the only reason you entered Ireland is to enter the UK illegally by crossing into Northern Ireland avoiding UK immigration border checks.
That is not to say, though, that any other country will not pay extra interest to you if they find out you have been refused entry to any other country, although I would expect it to cause less problems than in the UK and Ireland case.
Upvote:3
So yes, first and foremost, if you have ever been turned away or sent home from any country, you can expect more scrutiny at borders for a significant length of time after that. But second and probably just as important, "more scrutiny" is not "turned away." You report that the UK sent you back (it might be good to know if that was "we allow you to with draw your request to enter and go home" vs "we are excluding you", but whatever) and then you describe your experience entering Ireland as somehow the same.
This is important: it was not the same. You were allowed in. Yes, you had to promise not to try to sneak into the UK through the undefended and uncontrolled border between Ireland and Northern Ireland, or to fly across the Irish Sea which is also pretty uncontrolled. Yes, you had to promise to leave on the day you had always planned to leave. And you kept both those promises. They weren't even a burden to you. The few extra hours in the airport might have been a burden, but the experience as a whole was nothing like being sent back and not allowed in.
So, possibly, when you try to enter an unrelated country, they will know (from looking in your passport or their computer systems) about the long ago UK problem. But they will also then know about Ireland and how you kept your promises. Chances are, they will let you in, possibly after asking for similar promises from you again. The more you travel, the less important that long-ago UK interaction will be.