It’s not clear what specific language led to Sir Lindsay’s intervention.
A Conservative source claimed the Speaker was unhappy when Badenoch said Labour MPs “don’t like it up them”, a reference to the sitcom Dad’s Army.
But a Labour source claimed the Speaker’s comments related to Badenoch’s criticism of Phillipson.
The pair exchanged words in the division lobbies afterwards, with a Labour source claiming that Badenoch told Phillipson: “You are spiteful, I’m never going to stop talking about how spiteful you are.”
Phillipson is said to have replied: “The public are going to find out who you really are.”
A Conservative source said Badenoch also told Phillipson: “I will fight you all the way, you’re destroying children’s lives” – but that has subsequently denied by a source close to Phillipson.
Posting on social media, Phillipson later said “Kemi lost her head at PMQs – and afterwards too” and claimed the Conservative leader had previously compared her “to a Gestapo officer”.
She added: “I wonder what it is about a working class woman driving record investment in state schools by ending private schools’ tax breaks that the Tories hate so much.”
Earlier in the Commons, Sir Keir defended Phillipson and her “incredible story of social mobility and success” after growing up in poverty.
He said: “She knows that for poor children education is absolutely vital and that is why it drives every single priority and value that she has.”
Badenoch replied: “The fact is if she knew so much about poor children she wouldn’t have given them fewer teachers.”
The prime minister also staunchly defended other ministers, including Reeves, who he said “ended austerity” and had delivered the “right economic plan” to help “weather the global storms”, including the Iran conflict.
Sir Keir said he believes he will hand over the country in “better shape” than he found it.