Bi-generation
Bi-generation was a variant of regeneration which the Doctor believed to be a myth. Instead of one incarnation directly transforming into the next, bi-generation caused the new incarnation to split from the previous, allowing the two to become independent beings.
The Fourteenth Doctor experienced a bi-generation after being gravely wounded by a Galvanic Beam. This resulted in him continuing to exist as an incarnation distinct from the newly created Fifteenth Doctor, although this new Doctor's consciousness continued from the Fourteenth's after his rehab. The process was assisted by Donna Noble and Melanie Bush who each grabbed and pulled the Doctor's arms to separate the two incarnations who both remembered their whole lifetime prior to the bi-generation.
The Fifteenth Doctor later caused the TARDIS to undergo a bi-generation of its own using the fact that he had won a prize for defeating the Toymaker so that both Doctors had their own TARDISes. He was able to accomplish this as the Toymaker and his rules still lingered for a few moments after his defeat. (TV: The Giggle [+]Loading...["The Giggle (TV story)"])
Behind the scenes
In the in-vision commentary for The Giggle, Russell T Davies suggested, perhaps jokingly, that the bi-generation affected the Doctor's whole timeline so that previous incarnations may have continued to exist beyond their on-screen regenerations.
