How is the Rinne Sharingan awakened? I know Madara and Kaguya were the only people to do so but I don't remember it ever being explained in Naruto Shippuden.
To awaken the Rinne Sharingan I believe that you need to:
1.) possess the Rinnegan, and 2.) approach the moon/be near the moon.
Based on the Rinne Sharingan Naruto wiki page, it was written on the stone tablet handed down in the Uchiha clan that "when one who wields the power of Rinne approaches the moon, the eye which can reflect off the moon and grant the infinite dream will open". The eye here refers to the Rinne Sharingan, which Madara was able to acquire after approaching the moon.
You could read the rest on the wiki page about how Kaguya awakened it, if you would actually call that awakening. I'd like to believe that the logic behind approaching the moon to activate it was because the moon is essentially where Kaguya is, after being sealed by Hagoromo and Hamura.
*SPOILERS IF YOU HAVE NOT WATCHED THE ANIME UPTO DATE *
The Rinne Sharingan (輪廻写輪眼, Literally meaning: Saṃsāra Copy Wheel Eye) is a dōjutsu kekkei mōra and the predecessor to the Rinnegan and Sharingan.
I got the above from the naruto wikia page here.
From that we can conclude that Rinne-Sharingan is a predecessor of both Rinnegan and Sharingan.
From the anime, i know that in order to awaken the Rinne-Sharingan one must possess both the chakras of Indira and Ashura.
The fact that Hagoromo, father of these two had it proves the above line that both their chakra are needed.
Thus only Madara and Kaguya have the Rinne-Sharingan. Kaguya got it by eating the Chakra fruit from the God tree during the time of her pregnancy in order to save herself.
Madara got it when he was awakened with the cells of Hashirama infused in himself.
As a result of Black Zetsu's actions, Madara's body then served as a vessel for the resurrected Kaguya, who retained the Rinne Sharingan.
This is during the 4th war, when Kaguya emerges from the body of Madara. The above quote was also taken from the same wiki page.
I read the wiki of Madara, and got that while Madara was fighting, Black Zetsu used Obito to use Kamui and transport them to the real world. Then Madara took back his Rinnegan, giving Obito back his Mangekyou Sharingan. So Madara stalled Sasuke and Naruto with several Chibaku Tensei. So, Madara got into the proximity of the moon, during that time awakened the Rinne-Sharingan and succeeded in casting Infinite Tsukuyomi. I'd say to unlock the Rinne-Sharingan, you'd need the Rinnegan, basically the cells of Indra and Asura, and have the full clear view of the moon infront of you. If this is wrong, please correct me.
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To awaken the Rinne Sharingan, you have got to be the ten tails jinchuuriki.
Additionally, Sasuke's left eye is not a Rinne Sharingan but a Rinnegan given to him by the Sage of six paths, which is probably the reason for the tomoe.
We don't know if there are other variations of the Rinnegan, since we only see two pairs of them (Hagoromo's and Madara's since those we're the eyes Nagato and Óbito had).
I don't know why Obito didn't awaken his though, since he was the ten tails jinchuuriki for a while.