In recent weeks, when I look in the mirror, I have started to spot a half-resemblance between me and my grandfather. It could be that the large, thick-rimmed glasses he started wearing in the 1970s are not too dissimilar from my own – or maybe I’m simply getting older.

Or perhaps it’s just guilt. Grandad “married out” so he was the last “proper” Jew in the family. Under Orthodox Jewish law, Jewishness comes from your mother, so none of his children, let alone his grandchild, were Jewish, although we have retained a token observation of the major festivals – especially Passover. This year it runs from the evening of 30 March to 7 April.

On the first night of Passover, the youngest child present is charged with asking questions about the ceremony. As the sole grandchild in my family, that task would always fall to me, so the opening lines are clear in my mind: why is tonight not like other nights?

Increasingly, a version of that question is asked about Jeremy Corbyn. Why is this Labour leader not like other Labour leaders? For Corbyn’s most steadfast supporters (and many within the leader’s office) the answer is obvious: it’s because Jeremy Corbyn is the most left-wing politician ever to occupy the post of Labour leader, and therefore his political enemies are determined to block his path to government by any means possible. I’m reliably told that the suggestion he is anti-Semitic – or that Labour under him has become a more welcoming home for anti-Semitism – is one of the few topics that rouses Corbyn to genuine anger.

However, for many Jews there is another answer to the question of Corbyn’s difference from his predecessors. They point to actions such as his description of Raed Salah, a spreader of the “blood libel” that Jewish people drink the blood of Gentile children, as an “honoured citizen”. Corbyn once described Hamas, a paramilitary organisation whose charter contains anti-Semitic statements, as “friends”. Last month, in an incident that has brought Labour and anti-Semitism back to the front pages, it emerged that he had defended a mural depicting a series of anti-Semitic tropes.

It’s a sad truth – and one that pre-dates Corbyn’s leadership – that if you spend any time either participating in or covering left-wing politics you will be confronted by an expression of outright anti-Semitism. That doesn’t mean “a robust denunciation of the government of Israel”. More like “a suggestion that the Holocaust was faked, or that Jewish bankers run the world”. In some parts of the Labour right, tolerance for such tropes combines with Islamophobia – “I don’t like these anti-Semitic messages either, but I’ve got to get my Muslim vote out somehow” – while in others it travels alone. During the struggle for control within the activist group Momentum, many of those briefing against its leader Jon Lansman strayed regularly into anti-Semitic territory, frequently criticising his supposed links with Jewish journalists. (A Jew who talks to other Jews? Must be a conspiracy.)

After a while, those encounters begin to corrode the soul. Some days, my journalistic mask drops and I explain that while, yes, the government of Bibi Netanyahu is a roadblock to lasting peace in the Middle East, no, that can’t be blamed on the Jews of Stamford Hill. But mostly, of course, I don’t.

The resulting guilt is probably why my grandfather’s face haunts my mirror. And I know that the same uneasy dilemma plays itself out for many Jews who participate in, and not merely cover, left-wing politics. The fear that many – not all, but many – Jews have is that Corbyn’s long career in left-wing politics has, at best, seen him grow so accustomed to overlooking anti-Semitic remarks from his allies that he has become deaf to them. That is why he is not like other Labour leaders. Tony Blair and Gordon Brown, although they did not make a show of cracking down on anti-Semitism in the party, always defined themselves in opposition to the anti-imperialist, anti-capitalist and anti-Zionist section of the left. And Ed Miliband, who recognised Palestinian statehood and was highly critical of Israel’s 2014 war in Gaza, is himself Jewish.

Israel adds to the problem. It is not true to say that claims of anti-Semitism prevent legitimate criticism of the state of Israel. However, it is true that full-throated support for the state of Israel would reassure many Jews about Corbyn’s credentials. While a minority of British Jews still regard the creation of a Jewish state as a mistake (and many more are simply apathetic about its existence) others cherish the idea of a Jewish homeland. Knowing it exists, and knowing that politicians in their home countries support its existence, reassures them that they will never need to flee there.

How might Corbyn assuage those fears? He would always make an unconvincing cheerleader for the state of Israel even if he had the inclination to become one. But he is in an enviable position in that he enjoys unchallenged dominance in the party beyond that of any Labour leader since pre-Iraq Tony Blair. He has a comfortable majority on the party’s ruling National Executive Committee, his first choice as Labour general secretary has just started in the job, and a majority of members support him.

Corbyn could use that power to purge the party of anyone who engages in anti-Semitic behaviour. He could deliver a speech in which he lays out the difference between criticising Israel and being anti-Semitic: he could even draw on some of the material handed out at anti-war rallies by Jewdas, the organisation of radical left-wing Jews whose Passover gathering he attended on 2 April. He could, in short, prove that he is not like other Labour leaders who, either through weakness or lack of inclination, did not get the party’s house in order by showing zero tolerance of anti-Semitism.