It includes serious spoilers even from the explanation, so beware of spoilers.
This story is named 'Rezero EX : Repeating Life in a Different World from Zero', written on 2016.4.1.
It takes place 2 years after the Arc 3, deviating from Arc 4 chapter 74.
It is What-IF scenario of Subaru making a contract with the Witch of the Greed, Echidna's Soul, and taking different approach on the use of 'Return by Death'.
Edit : Forgot to write what the contract was about. Echidna basically wanted to see Subaru abuse 'Return by Death' even more, to fulfill her curiosity of what might happen in each loop (Hence the Greed). In the main story, Subaru refuses her offer because Satella requested him to cherish his own life even in the loop.
Edit2 : Added some dialogues in the story
Subaru wakes up in the morning, practices smiling in front of mirror (like in Anime Episode 8), and enters the dream-world of Echidna. First drinking tea as they always did, Subaru asks her what date today is, which is 14th of Kisdam. After they finished chit-chat (About if Echidna sees him while bathing or in toilet. Echidna gave vague answer of yes), Subaru returns to the reality.
[Subaru : You really stop looking at me if it is ethically problematic, do you?]
[Echidna : Of course, I am not that low to look at you while in bath or in toilet. But, I can't just look away if there is no assurance that nothing will happen in such defenseless situation. This is only to uphold the contract faithfully as Witch of Greed-]
[Subaru : I will take off the pendant while bathing or in toilet]
When Subaru returned, shortly Subaru's personal maid, Petra enters the room to wake Subaru (blonde girl in town with a ribbon, you could see her with Ram in Anime Episode 15). She makes some worrying about weather being cloudy, because she was scheduled to going shopping later. After greeting her, Subaru got out of his room to do his daily routine of meeting Beatrice, Rem and Emilia (Echidna continues to communicate him via telepathy)
While going to Beatrice, Subaru saw Garfiel(Arc 4 Character) and Reinhard training. It is revealed that Subaru repeated indefinite time of loop to know every villagers movement, and solved the incident in Arc 4 all by himself with Echidna's advice. Garfiel is hostile to Subaru because he was totally ignored in the solution, although persuaded by Ram and Frederica to join Emilia's side. Reinhard's mood isn't better neither. Because Felt forfeited royal selection and ran away, he lost his master and persuaded by Subaru to enter Emilia's side. It was blessing for Subaru, as Reinhard joining Emilia's faction brought some political advantage that the Sword-Saint of the Astrea, whose ancestor had sealed Satella in the past, took support of the half-elf.
Echidna praises Subaru's acting, because it was Subaru himself who persuaded and assisted Felt to run away from the royal selection. He had so many trial errors to make it succeed unnoticed, including killed by enraged Reinhard more than 20 times. Echidna justifies Subaru's action, saying it was only way to save more lives, which Subaru agrees.
Next, he goes into the fake library where Beatrice resides. After Roswaal's mansion was burnt off, Beatrice couldn't find anything to compensate for the destruction of Forbidden Archive, which broke her hard. Subaru tried countless times, but he ultimately choose to forcefully take Beatrice outside that time. From then, Beatrice became hikkikomori who never gets out of her room imitating the Archive.
Subaru enters the room calling Beako and got relieved when Beatrice said shut up, because he hasn't heard any words from her recently. She lost her meaning of life when she broke promise to her mother, and got furious on Subaru who already ditched her by dragging her out of the Archive. Subaru tries to comfort her by hugging, which Beatrice answered with scratching and bleeding Subaru's face and neck. Subaru get out of her room when she finally got silent and motionless.
[Beatrice : Betty is tired, gave up, I suppose. Failing my mother's order, breaking the contract, then still breathing, why?]
[Subaru : Beatrice...]
[Beatrice : If you just ditched me then... Why have you came to help? You're not 'Him' anyway!]
When Echidna said Beatrice's life was more important than the Archive, Subaru got angry if Echidna has any right to say that, because Echidna is Beatrice's creator and ordered Beatrice to take care of Forbidden Archive until giving it to 'Him'(Not specified). Echidna makes excuse that it wasn't intentionally done, and Beatrice could find other meaning of life someday if she lives on. Subaru thinks it as sophistry but agrees on anyway.
While Subaru is walking back, he meets Roswaal. Subaru knows him and Roswaal is using each other for their own goal, but doesn't know exact motive of Roswaal. Roswaal saw Subaru's wounds but says there are only 3 people who could use healing magic in the mansion (Beatrice, Garfiel, and Emilia) and he has only one choice considering the relationship. Roswaal then recommends what Subaru has done till now with the loop and encourages him to continue this trend (Roswaal knows Subaru can loop, but he doesn't know anything more than that, like how he loops or saves) While Subaru feels uncomfortable because he already suffered from Roswaal's betrayal and doesn't know his true goal, he could not leave Roswaal as he is too valuable.
(To remind, Rem lost her soul and existence in Arc 3, so Ram only heard from Subaru that unconscious Rem is supposed to be her sister)
When Subaru arrives in a room where Rem sleeps, Ram was already there. Ram became very polite to him and apologizes for what she have done to him in the past. When Ram apologizes about Rem taking Subaru's precious time away, Subaru stops her, feels weird how drastically Ram has changed from the past. Subaru asks Ram to leave the room before his awkwardness grows more, fearing what would happen with Ram when Rem awake.
[Ram : Again, I am really sorry for the rudeness in the past. Ram's ignorance has caused great inconvenience to Natsuki Subaru-sama. However, I am ordered to show Natsuki Subaru-sama greater respect than even Roswaal-sama or Emilia-sama, so it won't happen again]
Left with unconscious Rem, Subaru go over the ways to awake Rem from Echidna again. Echidna says there are 3 ways. First is to seek the Archbishop of Gluttony, who Subaru never found any trace of. Second is to seek the assistance of the Sage, but Echidna assures that Sage won't be of any help for Rem anyway. Third and only viable solution Echidna claims is to seek the Dragon's Blood, which grants miracle but could be obtained only by the Royalty.
Echidna really hates Emilia for unknown reason and undermine her importance as a mean to awake Rem. Subaru firmly reject that notion and says he is helping Emilia because it was his original goal, not as a mean for awaking Rem.
After circling his location of daily routine (Beatrice and Rem), he goes to his final destination, Emilia's room. When Subaru enters the room, Emilia saw the wounds on him and immediately concluded it was Beatrice's doing. Emilia couldn't hide her rage and froze everything near her unconsciously. Subaru took some effort and hugging to return Emilia to normal. After healing Subaru's wound, Emilia asks Subaru what to do next as she already finished studying Subaru ordered her to, which Subaru answers with a perfect smile he practiced in front of mirror in the morning. Then Emilia asks Subaru to pet her head for reward, and Subaru leaves the room sadly looking at Emilia's shortened hair (likely imitating Rem's appearance and behavior)
[Emilia : Subaru, you look hurt. What just happened?]
[Subaru : Ah, I just tripped-]
[Emilia : Liar]
[Emilia : Beatrice, isn't it?]
(Emilia exploding in a coldness)
[Subaru : Emilia! Emilia! Calm down! It's okay, okay!]
Emilia also broke hard when she couldn't pass the Trial herself (about accepting one's past). Instead, Subaru took care of everything without Emilia. Subaru first thought Emilia's over-dependence on him was temporary result of trust. When he found out it was more like severe mental disorder, it was too late because save point was already established after that.
Echidna doesn't hide her disgust at Emilia, how she became a doll to Subaru's convenience. Subaru tries to ignore what Echidna says. Echidna continues her sneering saying it would be a sight to see a queen who couldn't do anything on her own, always asking Subaru's confirmation. It brought Subaru's memory of Otto, who left Subaru saying "I considered you as a friend, but you never saw me as one, always trying to do everything on your own."
[Echidna : Always asking for your confirmation in the end, she would become really interesting queen]
[Subaru : Shut up]
[Echidna : You're also worrying, if it is okay to put 'that thing' on the throne. Always swaying, but you have to do it]
When Subaru returned to his own room, he met Elsa, who works as secret agent for him now, currently looking for the whereabouts of the Archbishop of Gluttony. Subaru asks her how the weather is outside. When Elsa replied there is no sign of rain, Subaru asks her to cut his head off painlessly. Elsa reconfirm his order(Free to do whatever she wants with his bowel, but never touch other residents) and swiftly cuts his head off (Subaru feeling sorry for Petra who has to clean the mess up)
[Elsa : Any last word?]
[Subaru : Do it painlessly]
[...]
[Subaru : And do whatever you want with my guts, but don't touch others. Anyway, Reinhard is he-]
(His head flying off)
After he returns by death, Subaru gets really nervous about the date, immediately entering the realm of Echidna. Echidna tells him this is the morning of 14th day of Kisdam. Subaru is relieved but gets tensed up, because new save point is the premonition for the event which might endanger his life, learnt from the previous experiences of countless loops he has done. Echidna asks what made him to deliberately loop this time, which Subaru doesn't answer yet.
Petra enters his room, and it is repetition of what he have done until Petra starts worrying about the weather. Subaru assures her that today's weather will not be a problem. Petra asks how he knows, and he answers "Because I just returned for this."
Scene turns to the Echidna drinking tea alone in her realm, assessing Subaru's mentality. Because Subaru doesn't see any value in his life, he focuses only on other's life too much, disregarding anything that is not directly related to the survival of his precious people. She just loves that stupidity and his dependence on her, and will continue to assist Subaru with her wisdom and love.
The story ends with Echidna asking herself "Why love always diminishes?"
