But Macedonia clearly liked the name, too, meaning that there's been a massive stand-off for nearly 30 years, which has seen many protests and conflict.
Matthew Nimetz said in a statement Tuesday he has "no doubt this agreement will lead to a period of enhanced relations between the two neighboring countries and especially between their people".
The following month they opened liaison offices in their respective capitals and a new Macedonian flag - with the controversial sun replaced - was raised for the first time at the UN.
U.N. Secretary-General Antonio Guterres praised the agreement as "a demonstration of leadership to the wider region and beyond" and hopes it will inspire others involved in drawn-out conflicts "to work towards negotiated settlements without further delay", spokesman Stephane Dujarric said.
People's nationality will be defined as "Macedonian/Citizens of the Republic of North Macedonia".
"I now call on both countries to finalise the agreement reached by the two leaders".
Greece had also vetoed its neighbor's bid to join North Atlantic Treaty Organisation and the EU.
But before that, he said the agreement will be put to parliament for ratification in order not to waste any time and to allow Greece to lift its blockade on Macedonia's potential membership of North Atlantic Treaty Organisation and the EU.
"There is no way back", Zaev said in a news conference. "Thanks to you, the impossible is becoming possible", he said.
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European Union foreign affairs chief Federica Mogherini and Johannes Hahn, commissioner for European Union enlargement, said they looked forward to accession negotiations beginning with Skopje in June.
The prime ministers of Greece and Macedonia say they have agreed on "Republic of Northern Macedonia" as the new name for the Balkan country, ending a bitter 27-year dispute that had prevented the former Yugoslav republic from joining global institutions such as North Atlantic Treaty Organisation.
Tsipras and Zaev spoke on the phone yesterday for an hour, and said they would continue their conversation today. He added that Macedonia "cannot and will not be able in the future to claim any connection with the ancient Greek civilization of Macedonia".
Greece is to ratify the deal in parliament after Macedonia has made the necessary changes to its constitution, Tsipras said. Both countries agreed that the new name would be used both internationally and bilaterally. He did not, however, spell out what the name will be.
Greece had objected to the name "Macedonia", fearing territorial claims on its eponymous northern region, as well as cultural appropriation of the Greek nature of Ancient Macedonia, a historically Greek province and important part of Ancient Greek history that has no direct connection between the modern citizens, who hail from a mixed Slavic heritage.
The dispute has poisoned relations between the two neighbors since Macedonia's independence from Yugoslavia in 1991.
Defense Minister Panos Kammenos, whose right-wing Independent Greeks party is Tsipras' government partner, said Tuesday he would oppose an agreement in a parliamentary vote.
After splitting from the former Yugoslavia, Macedonia was admitted into the United Nations in 1993 as the former Yugoslav Republic of Macedonia (FYROM).
Officials had earlier told Reuters a deal over the name was imminent.