Anybody else wondering just what or where the kamui dimension is? I know I am! I have arrived at 3 possibilities. I will share them in order starting with most likely and ending at least likely.
1) The moon. Yes, the moon. Almost everytime the manga puts obito in the spotlight beginning in kakashi gaiden and up through the current chapters, there is an illustrative focus on the moon. Kakashi gaiden gave us two key moments of obito developing as a character which kishi immediately followed up with an arbitrary panel drawing the moon. I say arbitrary because it served no purpose establishing setting /time of day. It was strictly an accompaniment image with obito's growth. Obito's mokuton rampage was shown to be the first instance of his mangekyo behaving in its trademark spiral fashion. When using his mokuton and mangekyo together in the dimensional slaughter of the mist ninjas, the climactic carnage shows his power spiraling towards the moon. I don't think for a second it was mere dramatic effect.
The sharingan is a literal shard of the rinnegan. The sharingan, being the result of the sage's division of power between his sons means all sharingan abilities and powers would not fall outside the deliberate scheme of the sage. The sage would also have had exposure or use of such a dimension. The doorway open to obito and kakashi would be open to the sage in other words. i doubt any sharingan ability would be devoid of a link to the what the sage also had. The sage sealed the jyubi's body in the moon, its sealing stone but transporting something of that magnitude requires a special technique, kamui is an excellent choice if the sharingan indeed retains techniques, albeit weaker or costly, of the rinnegan.
Obito is instrumental in the moon's eye plan. Madara was able to unlock the moon and recover jyubi 's body but to cast genjutsu from the moon, a way to get there or reach it could be necessary, kamui solves this. We haven't heard much about the sage and space-time tech but if kamui is prominent and deserving of minato's appraisal then rest assured it was the sage's very same technique.
2) an empty world of blank imagination awaiting an introduction of Yang energies to shape it into a real world. This means it is a step or two above a genjutsu space like madara used in lecturing obito or itachi's tsukiyomi, those were spaces strictly in the mind whereas the kamui dimension is blank space converged with the mind as linked through the mangekyo level of power. It is mystical existence lacking in outward circumstance. The sharingan possesses some of the tools needed in ultimate creation. The kamui dimension would be a manifestation of those tools at work. Also, depending on how much info madara and tobi have concealed, the kamui under rinnegan and/or jyubi power could possibly manage to transport everyone on the planet to a new world that so far only obito's sharingan can reach. Where genjutsu mixes with that and when i can't say atm. If indeed a blank space to be filled with creation then this places obito much higher in importance to the story, well beyond a pawn or supporting role who's time is nearly up. That ability is an ultimate power.
3) An imaginary space. It's strictly imagination, not quite a blank canvas mentioned above but still a degree more in substance than genjutsu space. Kamui merely links reality to imagination without shaping either one. The sharingan is heavily tied to imagination and limited solidification of illusion. Kamui is another unique way the sharingan can conduct a relationship with dream and reality.
In conclusion, whichever turns out to be the case will also determine the importance of obito remaining throughout the rest of the story. Kakashi's possession of this power does not conflict with any of the situations. It can mean kakashi's kamui is linked to obito's imagination or some complicated variation of how sharingan behaves when dealing with illusion and shared dreaming. I don't have the energy to explain properly. There just is logical explanation for kakashi to reach obito's imagination through their proven to be linked sharingan.
Phasing, it gets really interesting when trying to fit the "phasing" effect into any if the 3 scenarios. I wonder what effect signifies. Scenario 2 and 3 can suggest that his phasing is making "reality into illusion" by sending his body to imagination space. This ties to his grandstanding as an inconsequential nobody. In scenario 1, any speculation regarding phasing bases itself on a lack of evidence from the manga. It also implies the nature of certain facets in the sage's relationship with the jyubi. Having presence in two places at once.
Kamui is vital to the moon's eye plan for either the need to transport the entire world to a special physical location (moon) or to transport the entire world into "sharingan space". The next step ofc is to then implement the genjutsu.
I'll hold off elaborating further until i see if anyone cares or has interest in discussing.