“We love that song,” said Siobhan Donaghy after Flatline, the uplifting 2013 single she and fellow original Sugababes Mutya Buena and Keisha Buchanan released using the name MKS for legal reasons. “We had to fight to get it online. We had to fight for the name as well, which we never thought we’d have to – but we’re the Sugababes again.”

For a nostalgic pop package tour, this was a stirring phoenix-from-the-flames story. Following their cool but only modestly successful 2000 debut album One Touch, Donaghy left and was replaced by ex-Atomic Kitten member Heidi Range. Over a decade filled with fine pop singles, the group became a running joke about the impersonality of manufactured pop, with each singer eventually replaced until by 2010 no founders were left.

The original trio have been reunited for a decade, but they only won the legal right to use the name Sugababes in 2019, and this is their first tour since. It had the odd but positive energy of a long-awaited comeback show from three performers who have been together in public for years.

They deserved the validation, though. Dressed in dark trouser suits and leather overcoats, they appeared in silhouette against the neon bars of their band’s riser, giving a Saturday nightclub energy to the opening trio of Push the Button, Red Dress and Hole in the Head. Their upbeat songs were always their most consistent, and they bookended the set with two more, Round Round and Freak Like Me.

The central section of the show was more relaxed and conversational, showcasing stool-bound ballads including Too Lost in You, Ugly and Stronger, and a couple of other MKS songs in Today and Love Me Hard. That Donaghy originally sang on only five songs here, including the three MKS tracks and their 2019 cover of Sweet Female Attitude’s garage hit Flowers, is another irony.

The fifth and finest, though – their mighty debut hit Overload, complete with seated dance moves – demonstrated how perfectly suited this trio’s blend of vocal personalities always were. It’s good to have them back, properly.

Music review: Sugababes, Usher Hall, Edinburgh