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Speaking at a reception at Manchester Art Gallery, the minister said the countries had entered a “new phase” of their relationship following Britain’s departure from the European Union and the consulate would make a significant contribution to building a new partnership.
He said: “For me, the challenges we have faced in recent years have resulted in a new conversation, not always positive, in the context of the British-Irish relationship.
“We need to find a way of fixing that and we will.
“But, more importantly, we need to understand the breadth and the scale of the connections and the closeness of those relationships between Britain and Ireland and indeed Ireland and the north of England.”
Mr Burnham, who gave the minister a shirt signed by former Manchester United player Denis Irwin, from Cork, told him: “We’re so proud you’ve chosen Manchester, this is a little token from us.”
As well as football, the politicians discussed business opportunities for Ireland in the north west of England, including in the construction industry.
Mr Coveney also met Conservative MP Wendy Morton, the minister for Europe and Americas, and held a round table discussion with business leaders.
Among those attending the reception was Boyzone singer Shane Lynch, who now runs a bar in Cheshire.
Mr Coveney was also due to visit the Liverpool Irish Centre and meet Mayor of the Liverpool City Region Steve Rotheram during his visit.
Mr Rotheram and Mr Burnham are planning to visit Ireland next year as part of the bid to drive forward Irish partnerships.
The consulate, which has been operating since July, will represent Ireland’s interest in England’s North East, North West, Yorkshire and the Humber.