Edeleth

Edeleth is the non-binary ship between Edelgard and Byleth from the Fire Emblem fandom.

Pre-Academy
One night, Byleth and their father Jeralt rescue three young nobles, Edelgard, Dimitri, and Claude from bandits. Byleth nearly takes a mortal blow for Edelgard while trying to project her and is only saved by Sothis' ability to reverse time and have them redo their actions. After the bandit attack Jeralt and Byleth are then summoned to Garreg Mach Monastery, where Byleth is made a professor of the monastery's Officers Academy.

At the Academy Byleth has the choice to become the professor of one of the houses present, including the Black Eagles which Edelgard is the house leader of. Before choosing a house Edelgard will note the similarities between herself and Byleth, in that they both appear to be stand-offish.

Black Eagles Route
After Byleth joins the Black Eagles they can slowly become closer to Edelgard as she begins to open up about her past and her destain for the current system of nobility and its use of crests.

Later during the Ethereal Moon a ball is held and Byleth can meet Edelgard at the Goddess Tower, which has been fabled as good luck for couples. Edelgard will tell Byleth how the story surrounding the Goddess Tower is actual based on her father who met her mother there while he was attending the Academy.

If Byleth and Edelgard are close enough by the Pegasus Moon then Edelgard will ask for them to return home with her so that Byleth can witness an important ceremony. If they agree then they are able to see Edelgard become the new Emperor. Soon after the Black Eagles embark on a mission with Rhea to the Holy Tomb. Once there, it is revealed that Edelgard is the Flame Emperor and intends to take the crest stones within the tome and start a war against the church. Enraged, Rhea will demand that Byleth kill Edelgard. However, if Byleth was present for Edelgard's coronation then they will be have option to "protect Edelgard", siding with her against the Rhea and the church.

After the Black Eagles manage to escape Rhea and flee to a nearby provisional camp, Edelgard explains her plan to the Black Eagles. After a brief scene with Hubert, she asks to speak to Byleth alone. During this conversation she appears vulnerable and almost nervous, despite having her war on the church planned for a long time:

"No matter how much blood flows at my feet, I will not relent. We must break the bonds that the depraved church has placed on Fódlan. These sacrifices will allow us to create a future where we never need sacrifice again. It may seem contradictory, but it's the only way. Heh. Listen to me. I made up my mind long ago. Yet here I am, seeking your approval. Tell me the truth, my teacher. Are you happy with your decision to stand by my side. Unlike me, you can still walk away from all of this. This path leads to the death of the archbishop and the servants of the church. Can you live with that?"

Byleth responds that they chose this path as well, relieving Edelgard.

A battle at the Academy quickly ensues as Edelgard captures Garreg Mach Monastery. In the aftermath, Rhea transforms into her dragon form and buries Byleth under rubble as she escapes. Byleth is not found for five years.

Time Skip: Crimson Flower
After having "died" at the battle of Garreg Mach Monastery, Byleth is awoken by Sothis and returns to the academy to be reunited with their students. Edelgard is the first person Byleth sees upon their return and after momentarily being shocked that they are back, Edelgard quickly hugs them, telling Byleth that she was heart-broken after their disappearance and couldn't find them no matter how hard she searched, but never gave up hope that they would return. In the reunion with the rest of the Black Eagles, their classmates will bring up how much of an anchor Byleth was to Edelgard, and Caspar mentions that Byleth's disappearance broke her.

Byleth joins Edelgard in the war effort as part of the Black Eagle Strike Force. They continue to work together to take on the Alliance and The Kingdom, before going up against their ultimate goal: removing Rhea and the church from power. In the final battle, Rhea refuses Edelgard's call for surrender, and instead sets fire to Fhirdiad, forcing the Black Eagle Strike Force to attack and put a stop to her. Working together, Byleth and Edelgard are able to take down Rhea who is killed in the aftermath. However, Byleth collapses soon afterwards from their choice to defy the fate laid out for them, severing their connection to the Goddess. Edelgard tries to listen for Byleth's heartbeat but after hearing none, she begins to cry. As this happens the Crest Stone embedded in Byleth's heart dissolves, giving them a heartbeat for the first time. Upon hearing it Edelgard hugs Byleth once more, happy and relieved that they're alive.

Time Skip: Other routes
If Byleth sided with the church in the Holy Tomb, Edelgard will still be waiting for them at the Goddess Tower five years later. She will ask Byleth to join her side once again, and when Byleth declines they will engage in a duel.

In Silver Snow or Verdant Wind, a cutscene will be shown upon Edelgard's defeat in which a beaten, broken Edelgard tearfully begs Byleth to end her in order to end the war. Her last words are her wistfully telling Byleth she just wanted to walk with them, and afterwards Byleth is shown with a sad, regretful expression.

In Azure Moon, Edelgard transforms into a Hegemon Husk as a last desperate attempt to defeat the approaching Dimitri and the Blue Lions. If she attacks or is attacked by Byleth, however, she will state that she "grows weak" facing her former professor. After Dimitri kills her for good, both of them look back in regret at her fallen form before leaving the Imperial palace.

Post-War
After the war is won, Byleth can use the wedding ring their father gave them to propose to Edelgard. In their localized paired ending it is said that they "were sometimes spotted leaving the palace to privately enjoy the world they created. How they spent those precious moments, none but the two of them will ever know." In their original paired ending, it is said that "At times, the two left the palace to go on intimate honeymoon trips together. However, how they spent those moments loving each other is unknown to everyone but them."

Moments

 * In Chapter 10: Where the Goddess Dwells if Byleth is female then Edelgard will carry them back to camp after they faint, otherwise she will get Hubert to carry them.
 * As the story progresses and they get closer, Byleth will start teasing Edelgard who is the only lord character to receive this treatment.
 * One of Edelgard's voiced lines in Fire Emblem Heroes references her attempts to paint a portrait of Byleth.
 * While Edelgard is harsh towards Byleth as she tells them they can't mourn Jeralt forever, she smiles and welcomes them back when they finally begin to move forward.

Fanon
While Edelgard can be romanced by either a male of female Byleth, the ship tends to be the most popular when paired with a female Byleth. She is also one of the few characters to show to have explicitly romantic feelings towards Byleth in the game outside supports. It is strongly implied, or at least interpreted by fans, that she has feelings for Byleth regardless of the route taken.

On AO3, it is the most written femslash ship for the Three Houses fandom and the most written ship for Edelgard herself.

Some fans see the final battle's cutscene as a metaphor: defeating Rhea gave Byleth a proper heartbeat and returned them to their "human" form, restoring their hair and eyes back to their natural color. Likewise, having Byleth on her side helped Edelgard retain her humanity and reveal the more humane and selfless reasons for wanting to destroy the church.

Edeleth's biggest rival is Rhealeth (Byleth/Rhea), as Rhea fans more often than not hate Edelgard with a passion.

Fandom
FAN FICTION
 * Edelgard/Byleth tag on FanFiction.Net
 * Edelgard/Byleth tag on FanFiction.Net

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Trivia

 * Edelgard is one of the seven characters who can have a romantic S-Supports with both a male and a female Byleth.
 * Edelgard is the first lord, excluding avatars, that can explicitly romance characters of both genders.
 * Crimson Flower, Edelgard's path, is the only one where the song "The Edge of Dawn" does not play over the end credits. "The Edge of Dawn" is a tragic ballad song from Edelgard's point of view, of her sorrow and regret for what her path cost her in the end; in Crimson Flower, the song is not necessary because the player's choice to walk with Edelgard saved her from death.
 * Edelgard and Byleth were both altered by the Crest of Flames: Edelgard when Those Who Slither in the Dark experimented on her as a child, and Byleth to save their life when they were stillborn.
 * Edelgard is the only lord character to get a one-on-one talk with Byleth before the final battle, where they discuss the truth of Byleth's identity and origins and Edelgard's understanding that Byleth may cease to be once Rhea is defeated.
 * Aside from Dimitri, Byleth is the only person, who calls Edelgard by her nickname, "El".