BELFAST, July 8 (Reuters) - Britain's new Prime Minister Keir Starmer promised on Monday to secure an improved agreement with the European Union on post-Brexit trading rules and revamp the "botched deal" signed by former premier Boris Johnson.

Speaking in Belfast after talks with the leaders of Northern Ireland, where post-Brexit trade rules have dominated politics for years, Starmer said his new government would first need to implement changes under the current agreement to build trust with the European Union.

"We think we can get a better deal than the botched deal that Boris Johnson brought home and we will work on that," Starmer, who won a landslide victory last week, told reporters.

"We're not going to be able to get a better relationship unless we've demonstrated commitment to the relationship and the agreements that have already been put in place," he added.

Labour has ruled out rejoining the EU single market or customs union but has said it is possible to remove some trade barriers with the 27-nation bloc, which Britain left in 2020.

The largest pro-British party in Northern Ireland ended a boycott of the devolved assembly after tweaks to trading rules secured by former prime minister Rishi Sunak in February, but it has since called for more changes.

Asked about the prospect of a referendum on a United Ireland after Irish nationalists Sinn Fein became the province's largest party in parliament, Starmer said he would "act in accordance with the Good Friday Agreement," the 1998 peace deal that ended three decades of sectarian violence.

Under the deal, a referendum is at the discretion of the British government if "it appears likely" to the minister for Northern Ireland that a majority would favour cutting ties with London.

Starmer, who visited Edinburgh on Sunday, is to continue on a post-election tour of the four nations of the United Kingdom with a visit to Cardiff later on Monday. (Reporting by Amanda Ferguson and Padraic Halpin; Writing by Conor Humphries; Editing by Janet Lawrence)