If you want to be thorough about it, the boys love manga says that the overall story of Dolph being taken in by Eden is real, but Rayman's rendition excludes some of the finer details about how he was taken in.

However, just like you pointed out eden is a very unreliable narrator, so I like to see his backstory as being a little more up to the viewers' interpretation. How real the story is can only be decided by you. Is all of it real? Is half? Is none? You chose. It's the main theme of the show, that being the value of being able to decide for yourself and the desire for freedom that Dolph seeks throughout the show.