Claudeleth

Claudeleth is the non-binary ship between Claude and Byleth from the Fire Emblem fandom

Canon
Claude and Byleth first meet when Byleth saves Claude and the other houses leaders from an ambush from a mercenary hired to kill them. If Byleth has chosen to teach the Golden Deer House, Claude becomes extremely interested in Byleth's origins given Rhea's strange fixation with them along with the fact that they bear the Crest of Flames and can wield the Sword of the Creator. Claude's relationship with Byleth is flirtatious since the start. In the ball of the Monastery, Claude is the one who invites Byleth for a dance in the cut scene.

With the disappearance of Byleth, Claude never gave up hope that they were still alive. Remembering the promise the Golden Deer made, Claude returns to Garreg Mach Monastery and ascends the Goddess Tower. As he waits, Byleth, who went missing five years prior, miraculously returns. During the Verdant Wind phase, Claude becomes more proactive in pursuing his goal of a united Fodlan, and the final battle features a cutscene where he and Byleth take down Nemesis together.

In the proposal, Claude reveals that at first he wanted to use Byleth to accomplish his own goals, but he ended falling in love with her. They exchange engagement rings but live far away for some months, due to political affairs since Claude ascends to the throne of his kingdom and Byleth becomes the ruler of Fodlan. He eventually returns to be with her, as it's stated in the credits.

Fanon
The ship is one of the most popular romances of Fire Emblem. The pairing is just canonically possible if Byleth is female. The fact that Claude is not openly bisexual in game was a motive of frustration and great discussion in the fandom since he has a flirtatious streak with both genders of Byleth.

Claudeleth is seen as an equal partnership in which the two work together. This is emphasized by the aforementioned cutscene, and the fact that Claude is more well-adjusted than the other house leaders. At most, a Claude without Byleth never learns to follow through and take action, instead collapsing the Alliance and returning to his homeland of Almyra.

The pairing doesn't shy away from sadder or darker themes, however. Claude's backstory, while not as dramatic as Dimitri's or Edelgard's, is rife for hurt/comfort due to the racially-based bullying he received as a mixed-race heir to both a Fodlan noble family and Almyra's throne. Byleth's own origins provide fodder for drama, and "Female Byleth dies or almost dies in childbirth" plots are extremely popular.

Claude/Female Byleth is much more popular on AO3 than Claude/Male Byleth, but both versions get a good amount of representation in fanart.

Fandom
FAN FICTION

FAN VIDS
 * Comic Dub Fire Emblem Three Houses:The Night of the Ball
 * Fire Emblem 3H : Claude x Byleth AMV - Somewhere only we know
 * Fire Emblem Three Houses - Male Byleth and Claude (S Rank Support)