The SDLP has expressed its anger that drivers from Northern Ireland will have to carry an insurance 'green card' while driving across the border into the Republic.
The new rule will apply after the end of the Brexit transition period in January.
The Alliance Party said it was another negative result of the "Brexit shambles".
SDLP South Belfast MP Claire Hanna said: "This is yet another complexity and irritant resulting from Brexit.
"It's very ironic that Brexiteers campaigned for 'freedom from red tape', but the effect of four years of their terrible political decisions will be a long list of bureaucratic annoyances for businesses and consumers."
Alliance North Down MLA Andrew Muir said: "This is another outworking of the Brexit shambles that has to be addressed urgently.
"The UK Government must find a solution as part of the overall Brexit negotiations."
The Republic's Department of Transport confirmed the requirement to BBC Northern Ireland yesterday.
"A green card is a document issued by your insurer that proves you have motor insurance cover when driving in another jurisdiction," it said.
"As it stands, following the transition period, a green card will be required for vehicles from the UK, including Northern Ireland, being driven in Ireland or other EU member states, unless the European Commission declares otherwise."
The issue came to the fore during the Brexit negotiations.
The Republic's Department of Transport confirmed that it would not be an offence for vehicles from Northern Ireland or Britain to travel there without a GB sticker.
The green card may not be required if the matter is resolved in the UK and EU's post-Brexit deal.
According to the Association of British Insurers, drivers from the Republic crossing the border will not require a green card.
Talks resumed this week in London amid fears that time was running out for a deal to be reached before the transition period ends on December 31.
The Department for the Economy had previously estimated that there were approximately 110m border crossings in Ireland each year.
A green card is an internationally recognised insurance document that provides proof of the minimum compulsory motor insurance cover required by the country visited by the person driving the vehicle.
It provide a guarantee of insurance for a minimum of 15 days and can remain valid until the expiry date of the motor insurance policy, offering cover for multiple trips.
Currently all motor vehicles with a valid registration travelling within the European Union are covered by the terms of the EU Motor Insurance Directive (MID).