So, finished Oathbringer last night. I was talking with my husband, and he made an interesting point. When Dalinar reconstructed Honor's perpendicularity, it seemed to unite all three realms. Evidence: Stormlight doesn't originate in the cognitive realm (it manifests as a wave there too), but the perpendicularity released a huge amount of Stormlight.
Whatever that means, I want to focus on Odium's reaction. Odium/Rayse looked on in horror and said "No! We killed you!". Note that he said we, not I. Since he hasn't used the royal we anywhere else, that means he was talking about a group of which he was a part.
From this, I think he was looking at the Spiritual Realm and seeing Adonalsium, not Honor. Further evidence: whatever he saw made him shrink back in horror and lessen the Everstorm. The only thing that would be more powerful than a shard (save Harmony's two shards) is Adonalsium his/itself.
I think that, in the Spiritual Realm where everything is perfected, Adonalsium never died. To paraphrase Navani, if it was really God, it couldn't have been killed. I think that the shardbearers merely broke the part of Adonalsium that showed up in the Physical and Cognitive Realms.
What do you think, sirs?