An incarnation of the Doctor, in one of their "Timeless Child" incarnations, was contacted, tested and recruited into the Division by Solpado. Where the policy of the Time Lords was of "strict non-intervention in other worlds and times", the purpose of the Division was to intervene in times when it was necessary. Officially the Division did not exist nor had operatives, and its history was mostly redacted from the Matrix. Eventually, they thanked the Doctor for their service, then wiped their memory.
In the fictional story of Brendan in the Matrix, one of the clues to the truth was that the carriage clock he was given upon his retirement was dedicated for services to the Division instead of the Garda. (TV: The Timeless Children)
The Gallifreyan Gat and her recruited troop of Judoon were part of a force working for "the glory of Gallifrey", with Gat wearing a uniform similar to Solpado's. When the Fugitive Doctor tried to escape from service to their organisation, their job was to capture her and punish her for running away. (TV: Fugitive of the Judoon) Eventually, a Judoon cold case unit transmatted inside the Thirteenth Doctor's TARDIS and imprisoned her inside a maximum security facility for her crimes against the Division, (TV: The Timeless Children) as well as 7000 other crimes, though she eventually escaped. (TV: Revolution of the Daleks)
After learning the truth of her past with the Division, (TV: The Timeless Children) the Thirteenth Doctor sought out Karvanista for information on the organisation, as he was the only surviving Division agent she could find. He refused to tell her anything however. (TV: The Halloween Apocalypse)
Behind the scenes
- Though no link has been drawn in-story, the creation of a secret Gallifreyan organisation with authority to break the Time Lords' non-interference protocols, overseen by one of Gallifrey's Founders, is strikingly similar to the depiction of the origins of the Celestial Intervention Agency under the helm of Rassilon in such stories as The Legacy of Gallifrey. In turn, the Fugitive Doctor acting as a reluctant Division agent in a forgotten era of the Doctor's life in Fugitive of the Judoon is similar to the Second Doctor reluctantly becoming a CIA agent in the "Season 6B" era.