In the aftermath of Celta Vigo’s cancelled match with Real Madrid in February, when a storm ripped through the roof at their Balaídos stadium and accusations flew, their coach, Eduardo Berizzo, protested: “I still can’t make it rain.” No, some supporters suggested, but there is plenty you can do.

The team that beat Barcelona three years in a row, putting four past them the last two times they visited Vigo, had after all just deservedly knocked Madrid out of the Copa del Rey to reach the semi‑finals. Now they have reached a Europa League semi-final against Manchester United.

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“To do anything, you have to imagine it first,” Berizzo insists and he says his players have convinced him that Celta can “do something important” – but few imagined this. Although they were defeated in the cup by Alavés, denied the chance to face Barcelona in the final just as happened last year when they had defeated Atlético Madrid but fell to Sevilla, this is some achievement.

In June 2009, a late goal from a 21-year-old substitute rescued Celta from relegation to Spain’s regionalised third tier; on Thursday they host United. Iago Aspas, formerly of Liverpool, has been there before, but his team have not.

Celta have never won anything, although they have been beaten in three finals, and they certainly have never been in a European semi-final. Last week, they got there before anyone else, their meeting with Genk the only quarter-final that did not go to extra time. There were still six potential opponents as they started heading out of the stadium. “Whoever we face, we’ll go for them,” the centre-back Andreu Fontàs said.

It is very much their way. In the autumn of 2014, Celta beat Barcelona 1-0 at the Camp Nou, riding their luck a little, but the following year they hammered them 4-1 at Balaídos, and the year after defeated them 4-3. All over the pitch they suffocated Luis Enrique’s team, going for the throat. “When Barcelona don’t have the ball, they’re not Barcelona and every robbery was an automatic attack,” Berizzo said.

I want the ball the maximum amount of time possible and when I don’t have it I want it back. Then I want dynamism, spark Eduardo Berizzo

That line broadly sums up Celta’s style even if, as the season has gone on, it has been adapted, the coach talking about playing with “intelligence” and a touch of pragmatism sneaking in. When it comes to excitement, few teams in Spain have been better to watch and the idea has not changed significantly, even as they have lost key players: Michael Krohn-Dehli went to Sevilla, Augusto Fernández to Atlético and Nolito to Manchester City, while Fabián Orellana left in the winter in controversial circumstances, accused of disloyalty by his manager.

“I like us to control games, to be protagonists; I don’t like us to be dominated,” Berizzo says. Berizzo is a former Celta defender who played in a Copa del Rey final and describes his connection to the club as intense, his experience full of “extreme moments”: they reached a cup final and the Champions League, suffered relegation and got promoted again. He has been influenced by Marcelo Bielsa, having played under the Argentinian coach at Newell’s and then joined him on the coaching staff with Chile. “A great apprenticeship,” he calls it.

“I want the ball the maximum amount of time possible and when I don’t have it I want it back,” Berizzo says. “Then I want dynamism, mobility, spark, and I want us to use it well. I want us to be a team whose movements make it hard for the opposition to read us.”

Technically accomplished, Celta press high with intensity, running opponents down and then running at them. At the heart of it is Aspas – hyperactive, creative and decisive, scorer of 22 goals in 39 club games and called up to the Spain side, a goalscorer at Wembley in November. “He has found his place in the world,” Berizzo says.

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Theirs is a style that can at times leave them exposed and one that the coach admits makes huge physical demands on the players. Celta have played more games than anyone in Spain, too, forcing them to manage their minutes carefully and leave the league to one side. Every player in the squad has missed at least six league games, including the goalkeeper Sergio, and domestic football has been virtually ignored lately. Natural starters have not even been travelling, which helps to explain a run of five defeats in the last six and just three wins (and seven defeats) in 12.

Europe has occupied them instead. Three rounds in a row, they had to go away in the second leg, at Shakhtar Donetsk, FC Krasnodar and Genk, and each time they made it through.

It has not been easy and it is about to get far harder. Now comes a fourth second-leg trip: to Old Trafford. They have fallen at the last before but they have overcome greater obstacles than this, too. Five years ago they were in the Spanish second division; now they are in a European semi-final.

“There have been hard times, but now we’re extremely proud,” said Hugo Mallo, the club captain and a former youth-teamer born across the Ria de Vigo in Marín. Fontàs said: “We’ve made history. Now we have to keep dreaming.”