Labour activists must become “salespeople for socialism”, Rebecca Long-Bailey has said as the party’s leadership candidates clashed in their first hustings event of the campaign.

Long-Bailey, a frontrunner who is backed by allies of Jeremy Corbyn, said she was kept awake at night by the problems of inequality and the climate crisis but also the question of how to present socialism as mainstream.

“Our job is how we act as the salespeople for socialism. How do we show people that everything we believe in in this room is sensible, credible and will transform our economy?” the leftwing candidate said.

“Because what we are talking about is being done in other European countries and they are not calling it far-left or crazy. It’s what a civilised society expects. And I want to be the leader that gets the aspirational message through to people.”

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The other frontrunner, Sir Keir Starmer, said he was most troubled by the “frustrations of opposition”, as he made his case that he was the leader who could “change lives for the better for millions of people”.

“We need to rebuild, we need to unite, to be that effective opposition and find our way to victory. I want to lead not just our party but our country,” he said.

Lisa Nandy said she was worried about “where this country is going” and the “collapse of our public services”, calling on the party to “up its game and recover its ambition”.

Jess Phillips said she was terrified by the party “currently not reaching the people who desperately need us” and Emily Thornberry answered that she was kept awake by Labour needing to be a “fighting machine” in time for the next election.

The candidates appeared in Liverpool for the hustings and had each 40 seconds to answer a pre-selected question, which meant there was little scope for them to start debating with each other.

All five agreed on issues ranging from dealing with the climate crisis to the need to deal with antisemitism after the party’s devastating defeat.

However, there were a few moments of tension, as Phillips, the backbench MP and women’s rights campaigner, accused frontbench candidates of failing to speak up loudly enough on the issue of antisemitism.

“The Labour party needs a leader who has spoken out against antisemitism, and other forms of harassment in fact. When others were keeping quite and somebody who was in the room, struggling for an independent system – lots and lots of meetings – I have to say I don’t remember some of the people here being in that particular room or being in those particular fights,” she said.

Thornberry rejected that assessment, saying: “I tell you what, Jess. I have always been clear about it and I always will. It undermines our party and it undermines our soul. We should be kicking out antisemites in the same way Oswald Mosley was kicked out of Liverpool in 1937.”

Starmer subsequently hit back too, saying he had “spoken out on the radio, on the media” about the need for change as well as in shadow cabinet behind the scenes. As leader, he said he would be asking for a weekly report on his desk about progress in dealing with antisemitism complaints.

Phillips also took a dig at Long-Bailey for pledging to replace the House of Lords with an elected senate, suggesting this was not the language or one of the priorities of most people.

“We’ve got to talk in the language of people on the doorstep. No one talks about federalism or wanting this senate or that senate,” she said.

Starmer got one of the biggest rounds of applause from the Liverpool audience for saying he would not give an interview to the Sun newspaper during the campaign.

“The attacks on Jeremy Corbyn were terrible. They came back on us on the doorstep. They know why they do it,” he said. “It’s going to happen again and we need to cut through that.”

Long-Bailey also took a tough approach towards the media saying she had experienced her “fair share of smears in the last few weeks” and argued that “no amount of compromise with the Sun, the Daily Mail and Telegraph will help us”.

All five of the candidates said they would aim for unity in the party, arguing that divided parties do not win elections.

Calling for everyone to “reunify and rebuild”, Long-Bailey said: “We have to be honest that over the last four years we haven’t been united.” She said it was not acceptable for Corbyn supporters to have been branded members of a “cult” or those on the other side of the debate to be told to “F-off and join the Tories”.

Phillips said the “name-calling within the party has been horrendous” and called for Labour to build a “broad team”.

Thornberry said Labour members should stop fighting each other and concentrate on fighting the Tories, Starmer agreed there had been “too much division”, and Nandy said: “We raise the bar, we defend one another and we move forward together.”