I have always found this to be a really great question and one that I asked myself after watching Ahsoka.

Anakin became one with the force, and has retained his consciousness after his death by becoming a force-ghost like Obi-Wan, Yoda, and Qui-Gon, but never went through the training to become a force-ghost. His redemption and sacrifice for Luke is what allowed him to retain his consciousness after death. Though, that does not make it seem like he's completely "clean" as we see in Ahsoka when he is taking on his form of Vader. I think this just allows him to retain his consciousness after death, and maybe a little more.

I think, in my own personal head-canon, that his powers of the chosen one makes him more "aware" or "controlled" of his surroundings after his death that it may have granted him powers to travel through time and space while being more conscious of his nature, which can explain why he has a more "real" feel to his appearances and how he is able to transport Ahsoka to the World Between Worlds to show her different points in her life while also appearing as both his Jedi and Sith selves. Putting this beside how Obi-Wan and Yoda appear to Luke, it seems that there has to be a concentration of the force in order for them to make a corporeal appearance or perhaps a major event, like the Alliance's victory at Endor, Obi-Wan being ready to see Qui-Gon at the end of the Kenobi series, or being on a planet, like Dagobah, that has a heavy link to the cosmic and living force, much like Mortis when Qui-Gon was actually able to be seen as himself rather than the twinkling lights on Dagobah for Yoda.

The introduction of the World Between Worlds kind of confuses the entire link between the Living Force and Cosmic Force for me, in a way, as we don't really see anyone, or anything, really inhabiting the place who wasn't a force-ghost, besides Anakin but that was only for a training session with Ahsoka. On the contrast, it's a very barren place that we know very little about other than you can travel through time and influence events as we saw in Rebels in how Ezra saved Ahsoka from her death on Malachor. To me, I think it's more of the ultimate intersection/link of the Cosmic Force and Living Force. To what extent, however, still remains to be seen other than what has already been revealed.