Mary Beard and Arron Banks have both ordered the liver. “If we couldn’t talk like this, then bloody hell – we’d have lost a lot,” declares the author of SPQR: A History of Ancient Rome. “I’ve really enjoyed our conversation,” agrees the author of The Bad Boys of Brexit. “Arron and I have discovered that we can agree quite a lot on the symptoms of the current malaise,” summarises the history professor after a morning spent with the Brexit financier. “We disagree a bit on the underlying diagnosis, and we certainly fundamentally disagree on the treatment.”

If this was the movies, Mary and Arron would be two cops forced to work together by their sarge. You know the formula: one plays by the rules, the other’s a loose cannon, a cowboy. Together they have to investigate a case – and maybe just learn something about themselves along the way. As seems in part to have happened with Mary and Arron – albeit in a Cambridge restaurant as opposed to the Los Angeles underbelly of Lethal Weapon. We shall come to their various political accords shortly, as well as the areas on which they remain perhaps eternally divided. In the meantime, let’s slap the working title When In Rome on this one.

(Incidentally, I am here with them after Arron telephoned with an enticing invitation. To wit: “I’m going to Cambridge to have lunch with Mary Beard after we had bit of an argument on Twitter. And I thought, because you seem to like writing about Ukip so much, you might want to come along and cover it.” One has to relish these moments of perspective. What self-respecting writer wouldn’t jump at the chance to play Boswell to Arron Banks’s Dr Johnson?)

As for the original argument itself, hopefully you are sufficiently well adjusted for much of it to have passed you by. Twitter should be regarded as my mother regards London – a place where all sorts of very silly and probably rather ghastly things presumably do “happen”, but to which far too much attention has been paid by politicians and various other people who ought to know better.

Still, by way of background: back in December, Arron announced that “the Roman empire was effectively destroyed by immigration”, only for Mary to cut in and advise, “I think you all need to do a bit more reading in Roman history before telling us what caused the fall of Rome. Facts guys!” “Yes,” shot back Arron, “sacking Rome nothing to do with the downfall (eyes to sky)”. And so on.

A month later, I join them midway through their lunch. Has he had a chance to explain the fall of the Roman empire to you, Mary, or at least be offered a visiting chair? It must have been an honour to sit at his knee.

“I’ve been reliably informed,” chuckles Arron, “that the Roman empire didn’t fall till about the 1400s. I missed it by about 1400 years!”

“That’s it!” says Mary approvingly. “1453. Well done, Arron.” It also turns out the sack of Rome wasn’t one Breaking Point poster short of a perfect EU cautionary tale. “Well, as someone once said, it was the most polite destruction of a city ever,” explains Mary. “They were there for about three days and moved on …”

Still, it’s not all going Mary’s way. “You’re facts-based,” argues Arron, “Your book is absolutely brilliant – a forensic demolition of all these myths of Rome, which actually happened to be all the myths I loved. I want Romulus to be true, I want the Shakespeare ‘Et tu Brute’ to be true … Crossing the Rubicon – you say in the book it’s not even clear where the Rubicon was! You took that away from me!”

“But I’ve given you so much more,” laughs Mary.

“Yes,” says Arron, “but you’re suffering from the same romantic blanking of the facts with the EU. The facts blatantly face you that the EU is not working: southern Europe is ablaze with a currency that’s just wiping it out, you’ve got awful people like Geert Wilders in Holland, Marine Le Pen in France, and still the EU is showing no signs of changing. If not now, then when? Your vision of the EU is like mine of Rome – a dream. It’s what you want it to be, not what it is.”

On some things they warmly agree ... like Theresa May’s pending Brexit negotiation being a complete waste of time. Photograph: Alicia Canter for the Guardian

It is fair to say that Mary is more of an idealist than Arron. She believes supranational collaboration is demonstrably essential; he thinks it leads only to corruption and the flow of money from the poor to elites. She ridicules his unshakeable belief that government is exactly analogous to business. Arron argues that “Trump has appointed the brightest cabinet of any American government almost ever.”

“How do you measure that?” asks Mary. “That just sounds like Cambridge talk – people sitting around the table and saying ‘that’s the cleverest man, that’s the wittiest man’”.

But Arron is probably at his most discomfiting on the gathering darkness of Europe’s economics. When he demands, “What has your liberal consensus got you?”, he explicitly isn’t against the gains of progressivism. But he believes the failure to underpin them with economics that felt like they benefited more than an elite means it is now all under threat. “You bought those rights on the never-never,” he says. “Well if we did,” insists Mary, “it was still a good deal.”

“I wish that Arron had put his enormous energy and charm into reforming the EU,” she reflects at one point, “because he could have done such a lot.”

I’m not entirely convinced Arron missed his calling – but he is sportingly relaxed about her constantly identifying his “lefty side”. Discussing his encounters during the referendum campaign, he says, “The Labour MPs were infinitely nicer, better people than the Tories, who were all – almost to a man – pretty miserable, horrible politicians.”

After they have warmly agreed to renationalise the railways and the energy companies, draw the House of Lords by lot because it works perfectly well for juries, scrap Trident, and counter the mania for solving every problem with legislation, Mary concedes that the philosophical borders of Banksland “lie in a slightly different place to where I’d previously thought”. And perhaps surprisingly, Arron concurs with her entirely that parliament made a huge mistake not factoring a second vote into the referendum process in the event of a vote to leave. “But they didn’t, because they never expected to lose.” Theresa May’s pending negotiation is jointly judged a complete waste of time that will get no concessions. “I’m a bit bored of Brexit now,” remarks Arron.

Certainly, he is gravitating westward, ever closer to Donald Trump’s orbit. They have had their revolution in America, he explains, while he is still plotting ours. Brexit wasn’t it.

This week Arron hosts a huge party in Washington on the eve of the inauguration, to which he hopes all the big hitters will show up, and is keen to stress close and continuing ties with the incoming administration. “You saw the photo of us in Trump Tower right after he won. Boris didn’t even get into Trump Tower. Where was he?!”

In the coffee shop opposite? Failing to get a booking at the Trump Grill? “Trump is actually so different to his public persona. We spent an hour with him and he was actually quite shy and modest. He was listening. Most of the politicians I’ve met are on transmit, not on receive. He was having a conversation. Then you put a TV camera on him and it’s like – boosh!”

Mary is unconvinced. “But his Twitter persona is non-negotiatory.”

“And that,” judges Arron, “is why he is going to be a great president.”

Like I said, there were one or two areas on which common ground between our two principals did not immediately suggest itself. Don’t even get Mary started on Trump’s pussy-grabbing tape, which she regards as several thousand orders of magnitude more telling than Arron does: “As Nigel said, you put two men in a locker room … well, they’re very silly, aren’t they?”

“My fantasy is going into a men’s loo,” muses Mary. “And listening to what they say.”

“Nothing happens in there!” hoots Arron. It’s totally silent!”

“I think that’s where they fix all the business,” she twinkles. “I think they go out of the meetings in my faculty, go to the loo, say they’re going for a slash, and then they come back and it’s all done.”

This feels like the moment for Arron to point out that his daughter is “a ferocious Remainer – and she’s got the whole political correctness thing going as well, and I get the thick end of it from time to time. The way I talk, the way I behave, the unfiltered nature …” (Furthermore, one of his daughters has a first in history from Exeter, which has led to various Beard-related lectures to her father.)

Even so, both think politicians must end their reliance on endlessly-repeated political platitudes or face futher rebellion. “It’s just complete nonsense,” says Arron. “The big society has now morphed into the shared society – what does any of it mean?”

“Fifty years ago people were happy to be talked down to,” says Mary. “And if there has been any success of the British education system, then it’s that people no longer feel that they’re going to be talked crap to by people who just happen to be in positions of power.”

“That’s completely true,” says Arron, who thinks that “politicians have taught poor people how to behave but not rich people”. Mary wouldn’t go that far. “People have become more rightly unsatisfiable,” she says. “My problem is what that impulse results in, and if it’s manipulated.”

Which brings her to the Breaking Point poster. “Was that the way to conduct the debate?”

The Breaking Point poster ... ‘Was that the way to conduct the debate?’ asks Beard. Photograph: Philip Toscano/PA

“We discussed it at length,” says Arron, who claims only 8% of the country were offended by it. “This was not done on the hoof. We played to win – we weren’t going to play Queensberry rules.”

“I prefer that argument to a lot of the other arguments,” Mary judges. “I think that sometimes there are too great consequences to winning, but at least that’s straight.”

He’s about to chip in, but she won’t let him. “Come on,” she pleads. “I’m about to concede something! As in Rome, it is always difficult to distinguish between invaders, economic migrants and refugees. And I think your Breaking Point poster was an interesting case. That poster is evidence – it will be in all the history books in 300 or 400 years time” – Arron looks rather pleased – “which of the three are these?”

“It’s not an invading army,” he says, “but they’re all young men and about 25 years old. If I was them I’d do the same thing.”

“Yes, but refugees mostly are young men,” Mary points out. “We are sold the idea of a refugee as a tiny child sitting crying, as a way of raising money, but elderly ladies and kids largely can’t move. The demographic is mostly young men.”

“I don’t hold that against them because I would do the same thing,” says Arron. “But it’s how many people you can take in and maintain cohesion, without it breaking down society.”

“Clearly there is risk,” Mary replies. “Young men on the move – there will be some trouble. But I am prepared to take that risk. And that’s a moral choice.”

They could go on for hours – indeed, they have gone on for hours. At least they’re talking. “Yes! I’m trying to get Arron to change some of his ideas for this political movement. Although I gave him a terrible idea about currency. Ill-advised, because I’m sure I shall hear it coming up again. I did point out to him that one of the major problems in the 5th century BC was when good old democratic Athens has its empire, and what does it do? It imposes a single currency. Because currency is all about power and control.” A guilty giggle. For his part, Arron insists he is actively drawing on Mary’s brain for his future plans. Where they go, might others follow? Arron doesn’t see why not: “I always try to be good-tempered with people.”

“Yes,” agrees Mary, “but if you look at your supporters, they don’t follow those rules.”

“Nor do yours,” Arron reminds her.

Still, you’ve got to start somewhere. I haven’t checked this with Mary – or indeed Arron – but am given to understand that Rome wasn’t built in a day.