A habit I developed in Asia when I worked out there was to carry my passport with me at all times. In some places I worked, technically you were obliged to. Not once in eight years did I have to produce it for a police officer.
That said it came in handy when cashing large sums of money etc, so it's a habit I've continued to this day.
Re Ireland, I've been to Northern Ireland quite a few times and crossing the border in either direction was a doddle with border checks only infrequently made (exclusively during the Troubles).
The area around Strabane, for example, is littered with small roads that were never properly checked during the Troubles and certainly won't be now. This led to a rather novel VAT scam which some of the locals got up to -- but that's a different story!!