How likely is France to wake up on Monday morning to a new far-right dawn?

That was the garishly painted, hotly debated scenario in media headlines, the EU in Brussels and seats of government across Europe following the first round of France’s parliamentary vote last week.

But despite the spectacular showing by Marine Le Pen’s National Rally (RN) party, the short answer is: an RN majority is possible. Not probable.

French centrist and leftist parties have strategically withdrawn candidates to bolster each other’s contenders ahead of Sunday’s decisive second round.

But the impact of this election will be seismic, whether or not the RN wins an outright majority – or whether Jordan Bardella, its social media-savvy young president, becomes France’s new prime minister.

Polls predict RN is all but guaranteed to win more seats than any other political grouping.

That means a decades-old taboo will have been shattered in France, a core EU nation.

The EU was born out of the ashes of World War Two. It was originally designed as a peace project, with wartime enemies, France and Germany, at its core.

Far-right parties were banished to the outer fringes of European politics.

Last month, world leaders gathered in northern France to mark 80 years since D-Day, the allied amphibious assault in Normandy that helped secure the defeat of Nazi Germany.

But now, “far-right” or “hard-right” or “populist nationalist” parties are part of coalition governments in a number of EU countries, including the Netherlands, Italy and Finland.

There are challenges in labelling these parties. Their policies frequently change. They also vary from country to country.

And their normalisation is not an entirely new phenomenon. Former Italian prime minister, Silvio Berlusconi, a centre-right politician, was the first EU leader to take the plunge. He formed a government with the post-fascist political group, Movimento Sociale Italiano, back in 1994.

Six years later, Austria’s conservatives went into coalition with the far-right Freedom Party. At the time, the EU was so outraged that it blocked official bilateral contacts with Austria for several months.

Post-war political etiquette dictated the political mainstream must form a cordon sanitaire, a “health barrier”, at election time to keep the extreme right out of European governments.

The universally recognised term for that practice is French, which gives you a sense how passionately many in France felt about it.

In the 2002 Presidential election, some French voters clipped a clothes peg to their noses on their way to polling stations – a way of showing they’d vote for a candidate they didn’t really like, just to keep out the far right.

This was a far right that for years was led by Marine Le Pen’s father, with French former members of a Nazi-led Waffen SS unit in his party ranks.

Fast-forward to 2024, and Marine Le Pen’s ambition, 10 years in the making, to detoxify her father’s party – changing its name and trying hard to clean up its image – appears to have been a roaring success.

The cordon sanitaire now has a searing gash in it, after the leader of France’s centre-right Les Républicains struck a deal with the RN not to compete against each other this Sunday in specific constituencies. This was an earthquake in French politics.

Crucially for Marine Le Pen, those who support her aren’t embarrassed to admit it any more. The RN is no longer viewed as an extremist protest movement. For many, it offers a credible political programme, whatever its detractors claim.

French voters trust the RN more than any other party to manage their economy and (currently poor) public finances, according to an Ipsos poll for the Financial Times newspaper. This is despite the party’s lack of government experience and its largely unfunded tax-cutting and spending plans.

Which begs the question, when you observe the angst-ridden despair in liberal circles in Europe at the growing success of the so-called “New Right”: if traditional lawmakers had served their electorates better, perhaps there’d be less of an opening for European populists to walk into?

By populists, I mean politicians like Ms Le Pen who claim to listen to and speak on behalf of “ordinary people”, defending them against “the establishment”.

This “them and us” argument is extremely effective when voters feel anxious and ignored by governing powers. Just look at Donald Trump in the US, the sudden unexpected breakthrough of Reform UK in Thursday’s UK election and the huge success of Germany’s controversial anti-migration AfD party.

In France, many perceive President Macron – a former merchant banker – as arrogant, privileged and remote from the everyday cares of ordinary people outside the Paris bubble. A man who made difficult lives even tougher, they say, by raising the national pension age and trying to put up fuel prices, citing environmental concerns.

It must be a source of frustration for France’s president that his success at lowering unemployment rates and the billions of euros he spent trying to soften the economic effects of the Covid and energy crises seem largely forgotten.

Meanwhile, the RN concentrated much of its campaign on the cost-of-living crisis.

The party has pledged to cut taxes on gas and electricity and to raise the minimum wage for low earners.

Priorities like these mean the RN should no longer be labelled a far-right movement, its supporters insist. They point to a widening support base and say the party shouldn’t be forever tarnished by its racist roots under Le Pen senior.

A similar argument echoes out of Rome. Italy’s Prime Minister, Giorgia Meloni, once used to praise fascist dictator Benito Mussolini. Her Brothers of Italy party has post-fascist roots but she now heads one of the EU’s most stable governments.

She recently censured a meeting of her party’s youth wing. Members had been filmed giving fascist salutes. There was no room in her party for nostalgia for the totalitarian regimes of the 20th Century, she said.

While critics at home warn of attempts to influence Italy’s media landscape and Ms Meloni’s attacks on LGBTQ+ rights, her concrete proposals to tackle irregular migration have won plaudits from the European mainstream, including the EU Commission chief, Ursula von der Leyen, and the UK’s recently ousted prime minister, Rishi Sunak.

Frankly, on hot-button issues like migration, it’s becoming increasingly difficult to distinguish between the political rhetoric of the far right in Europe and traditional mainstream politicians intentionally sharpening their speeches to try to hold on to voters.

Former Dutch prime minister Mark Rutte was a prime example of this, and Emmanuel Macron too, the more he’s felt the heat of Marine Le Pen’s popularity.

One of the inadvertent effects of mainstream politicians aping parties further to their right on migration is that it makes the original anti-immigration parties seem more respectable, acceptable and electable.

Witness the recent stellar performance in the Netherlands’ general election of anti-migration politician Geert Wilders, who has been regularly accused of hate speech.

The label “far right” is one that needs to be debated. Much depends on the make-up of each party.

But the kind of acceptance now enjoyed by Ms Meloni in wider international circles is still a remote dream for Ms Le Pen.

The RN insists a parliamentary majority is still within reach this Sunday. More likely, polls suggest, is a paralysed hung parliament or an unruly coalition government of non-Le Pen parties.

Any and all of these scenarios reduce Emmanuel Macron to a pretty lame-duck president.

Political instability at home means big EU powers, France and also Germany, are turning inwards at a time of great global uncertainty.

Wars rage in Gaza and Ukraine. EU and Nato-sceptic Donald Trump is poised to possibly return to the White House.

It’s a precarious moment for Europe to be without leadership. Voters feel exposed.

Even if not this Sunday, Marine Le Pen’s followers firmly believe their time is coming. Soon.

BBC InDepth is the new home on the website and app for the best analysis and expertise from our top journalists. Under a distinctive new brand, we’ll bring you fresh perspectives that challenge assumptions, and deep reporting on the biggest issues to help you make sense of a complex world. And we’ll be showcasing thought-provoking content from across BBC Sounds and iPlayer too. We’re starting small but thinking big, and we want to know what you think – you can send us your feedback by clicking on the button below.