How did Obito control control the fourth Mizukage with his right eye.

[a banned cartoon]

First off you are assuming that is Obito controling Yondaime, which i personally doubt.
Second off no one ever said the left/right were ninjutsu/genjutsu

i mean kakashi has left eye and its certainly not genjutsu

Actually in the anime and manga as well Zetsu states that either the right eye is for ninjutsu and left for genjutsu, or maybe he stated the right eye was for the ultimate offense, and left for the ultimate defense, but anyways I'm a 100% certain I've heard that on the anime or manga one.

 

[a banned cartoon]

Obito's true power came from his powerful right eye. It's only a theory, but I either think:

1) Obito is closely related to Shisui, and has a strong affinity with his genjutsu

2) Itachi gave Obito a single usage of Shisui's eye in exchange for his help on the Uchiha massacre

A small panel just to strengthen the idea a bit:

Spoiler

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Note what the Mizukage says.

Yes but that guy was using the byukugan

 

[a banned cartoon]

No.....
Itachi would NEVER give madara shisui's eye or any "usage"

Shisui entrusted Itachi with it, he wouldn't abandon it like that

Rather i believe madara somehow got Danzo's eye and danzo retrieves it later

Shisui's eye was implanted into Itachi's crow that he put into Naruto. Remember

 

[TheWillofFire]

Shisui's eye was implanted into Itachi's crow that he put into Naruto. Remember

Yea which is why i said danzos eye not itachis

 

[Makoto]

Obito's MS power is Kamui, that doesn't mean his sharingan can't cast genjutsu. Any sharingan can cast genjutsu, it's just that MS eyes have unique powers aside from genjutsu, coyping techniques, and reading your opponent's movements.