The Common Travel Area (henceforth CTA) consists of the UK, Guernsey, Jersey, the Isle of Man and Ireland.
Between the UK, Guernsey, Jersey and the Isle of Man, the border is fully open with no immigration checks. In addition, Guernsey, Jersey and the Isle of Man apply UK entry regulations and issue UK entry stamps in passports. Consequently, the UK is henceforth taken to include these territories.
With Ireland the situation becomes more complicated, as Ireland is not fully compliant with the open-borders concept of the CTA.
Specifically, non-EU/Schengen citizens are to receive an Irish entry stamp, even if arriving from the UK. For visa-free nationals, this lets them stay in Ireland for max 3 months, while for Irish visa holders the conditions of the visa apply.
On the contrary, those arriving in the UK from Ireland are not normally checked and are not to receive a UK entry stamp. Instead, non-visa nationals who entered the CTA through Ireland are allowed to stay in the whole CTA (incl. the UK) for the period authorised on entry to Ireland (usually 3 months), while UK visa holders are subject to the conditions of the visa regarding their stay in the UK.
If entering the CTA through Ireland, non-EU/Schengen citizens receive an Irish entry stamp. If also holding a UK visa (or being exempt) they can then continue on to the UK without any formalities. Though spot checks happen at UK air/seaports and the Irish land border, no UK entry stamp is issued, nor is one required.
On the downside, this means the time those without a UK visa are allowed in Ireland also includes the rest of the CTA. For travellers visa-exempt for both the UK and Ireland, the visa exemption is 6 months in the UK but only 3 months in Ireland, meaning if e.g. a US-citizen visitor enters the CTA through Ireland, they’ll only be allowed 3 months in the whole CTA.
Legal residents of Ireland are conferred a 3-month deemed leave when entering the UK from Ireland.
At Irish airports, all international arrivals will clear Irish border control, so passengers arriving from the UK needing an entry stamp will receive one with no hassle.
When arriving by sea from the UK, border checks are often carried out, but not universally. If there isn’t a check, pedestrians and car passengers should seek out the INIS (in Dublin) or Garda (in Rosslare) to receive an entry stamp. If travelling by bus, you should contact border control in advance (for Dublin, BMU@Justice.ie; for Rosslare, +353 539 133 204), inform them of what bus you’ll be on and that you need an entry stamp.
By land from Northern Ireland
As stated on the INIS website, non-EU/Schengen citizens entering Ireland via the land border have to visit the INIS (in Dublin) or local Garda station (elsewhere) as soon as possible to obtain an entry stamp.
It is not known to the undersigned whether all Garda stations handle foreign entrants, or only dedicated ones – this is being verified with the Garda.