When did Obito awaken the Eternal Mangekyou Sharingan? Spoilers if you didn't see episode 473 of the anime yet:


   In anime episode 473, Kakashi uses Obito's eyes to create Susanoo. But Obito's eyes were not Eternal Mangekyou. Previously, he had Rinnegan which he has borrowed in addition to one of his own eyes, and that's why he could do lots of stuff. 


Note: this is not the same question as the closure says! Eternal Mangekyou is when someone from Uchiha gives his Mangekyou to to someone dear then it becomes eternal and the powers will be activated.
Obito never had the Eternal Mangekyō Sharingan. The user does not need the Eternal Mangekyō Sharingan to use Susanoo according to the wikia - just the Mangekyō Sharingan in both eyes. Once Kakashi gains the power of both of Obito's eyes, it allows him to use Susanoo. Eternal Mangekyō Sharingan increases the power of their abilities and causes the user no negative side effects such as losing vision and bleeding from the eye.

The full body Susano'o is the most powerful form granted by the Mangekyo Sharingan. Kakashi said in episode 437 at 12:53 "Gaining the Six Path's power enhances one's Visual Prowess too". It's most likely that Kakashi was able to use the full body Susano'o since when he gained the pair of his (actually Obito's) Sharingan, he also gained some of the Six Path's power Obito posessed.