A very interesting question. The border between Northern Ireland and the Republic is non-existent across much of its length, many roads not even having a sign to indicate where the border is. There may be signs about speed limits (mph in Northern Ireland, kph in the Republic), and signposts in the Republic give the English and Irish placenames, but these could be a few hundred metres away from the actual border, so it would be surprising if there weren’t occasional accidental incursions.
However, as both are EU members, PSNI operations in the Republic are covered by the EU spirit of Mutual Cooperation, as are Garda Síochána operations in Northern Ireland. This is underscored by the European Council Framework decision on ‘The Simplifying and Exchange of information and intelligence between law enforcement authorities for the EU’ (18th December 2006) and the Crime (International Cooperation) Act (2003). There are also laws on police work outside jurisdiction specific to Northern Ireland ( Criminal Law Act 1975) and the Republic (Criminal Law Jurisdiction Act 1976).
Note that, typically, outside jurisdiction actions are normally permissible for a specific set of offences, and may be subject to approval by the police and judiciary of the operational jurisdiction. However, the level of approval needed is determined by the individual case, and can be formal or informal as required. Similarly, the police operations carried out are not limited, and may be determined on a case by case basis. Typically, the reason for the approvals/notifications is so that police from the operational jurisdiction can accompany police who are out of their jurisdiction, as is normal between any two police forces worldwide, especially where the crime can be prosecuted in either jurisdiction.
Thanks for that – I learned something new in checking this out!