The Christmas Inversion (short story)

This story essentially takes place at the same time as The Christmas Invasion, with the Third Doctor arriving in Jackie's flat after the Tenth Doctor's TARDIS has been taken up to the Sycorax ship and leaving after his future self has been returned to London.
The Third Doctor is unfamiliar with the Slitheen.

Notes

It's explained that the Tenth Doctor was able to complete his regeneration and stabilise his new incarnation by siphoning regenerative energy off of the Third Doctor, the latter referring to it as "energy osmosis".

Continuity

Jackie mentions the Slitheen, the pig crashing into Big Ben, and Harriet Jones helping save the world. (TV: Aliens of London/World War Three)
The Doctor mentions the "Omega business". (TV: The Three Doctors)
Jackie believes that the Doctor has regenerated again, and, when meeting Jo and Yates, she thinks that Rose and Mickey have also changed their appearance. (TV: The Christmas Invasion)
The Third Doctor is unfamiliar with the Slitheen. Despite this, his previous incarnation was familiar with Raxacoricofallapatorians in the form of the Hanazeen-Blathereen family. (PROSE: A Comedy of Terrors) When the Ninth Doctor confronts the Slitheen, he is able to recall their home planet only after narrowing it down according to their distinctive traits. (TV: Aliens of London/World War Three)
UNIT's secret HQ is no longer located inside a building marked with a sign which reads "UNIT Headquarters," (TV: The Five Doctors) and is instead in a base under the Tower of London. (TV: The Power of Three)
Jackie asks Jo why she isn't in the Doctor's future, and wonders if he makes a habit out of taking people with him before moving on. Rose herself will confront this thinking when meeting Sarah Jane Smith. (TV: School Reunion)
Jackie recalls that Rose went back to "certain death" to save the Ninth Doctor. (TV: The Parting of the Ways)
Jackie recalls Rose telling her that she had met her husband and her younger self. (TV: Father's Day)
The Doctor claims to have "never known a Rose". According to another account, his granddaughter was originally named Arkytior, "rose" in High Gallifreyan. (PROSE: Roses)
When Mike Yates mentions UNIT's name (United Nations Intelligence Taskforce), the Beefeater notes that it's actually "Unified Intelligence [Taskforce]." This same discrepancy was noted in a conversation between Jack Harkness and John Benton. (COMIC: Official Secrets)
Jack Harkness later mentions Rose when he inquires as to how Jo left the Doctor's company. (AUDIO: The Green Life)