Froome lost 22 seconds to stage victor Romain Bardet of France at the end of a short, steep climb in Peyragudes, and 20 to Italian Fabio Aru, who snatched the overall leader's yellow jersey from the defending champion.
Aru's overall lead going into Friday's 13th stage, 101km from Saint Girons to Foix in the Pyrenees, was 6s on second-placed Froome, 25s on Bardet in third, 56s on fourth-placed Uran and 1 minute 41s on Ireland's Dan Martin (Quick-Step Floors) who was sixth on the stage at 13s.
Sunweb rider Barguil won in a sprint finish from a four-man breakaway in Foix, beating Nairo Quintana, Alberto Contador and Team Sky's Mikel Landa. In addition to the stage win, he's got a comfortable lead in the KOM competition with all the points he soaked up today.
"Yesterday (Thursday) was a hard day for me but today I felt a lot better and we played a good strategic game with our team". With Froome only six seconds behind him overall, the leader of the Astana team knows he can't afford to let the Briton get away from him.
Landa was the biggest victor of the day, jumping from seventh to fifth, now 1:09 back.
It is the first time in his career that Froome has lost the yellow jersey mid-race, having carried it all the way to Paris once he had taken it in each of his three Tour wins to date.
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On Bastille Day, it was appropriate that a Frenchman should claim victory in the stage, and that was the case for Warren Barguil.
Vinokourov, though, said this stage was significant, not least because Froome has often excelled in the Pyrenees in the past.
"This shows that Froome is beatable", exhorted Alexander Vinokourov, Aru's Astana team manager.
"Yesterday was a hard day for me but today I felt a lot better and we played a lot with our team", said Froome. Sure, it's been 32 years now since they last won their home Tour.
Landa is now fifth, 1:09 behind Aru, and Team Sky's management will need to make clear who the leader is after the Spaniard, whose contract expires at the end of the year, finished ahead of Froome on Thursday. Later in his post-stage interview, Froome regretted giving the Italian the freedom to attack and that mistake must be haunting him all the more after Thursday.