Ineffable Bureaucracy

Ineffable Bureaucracy is the non-binary ship between Archangel Gabriel and Beelzebub from the Good Omens fandom.

Good Omens (2019)
Gabriel and Beelzebub are representatives of Heaven and Hell respectively. Gabriel is the first Archangel, so he's beneath Metatron and God Herself. Beelzebub is the Prince of Hell and Lord of the Flies, so ze is the second-in-command to Lucifer. Throughout the series, both Heaven and Hell prepared for the Apocalypse as it would be set up by the time the Antichrist Adam Young would turn eleven. Both of them want the apocalypse, as they see it as a competition for which side is "better".

Both the Archangels and the Demon Lords noticed that the angel Aziraphale and the demon Crowley accidentally watched over a regular boy for eleven years and that they spend an awful lot of time together for someone who should hate each other's guts.

After Adam Young refused to start the Apocalypse and managed to defeat three of the Horsepersons with the help of his friends, Gabriel and Beelzebub appeared in person and tried their hardest to appeal to him in hopes of starting another apocalypse. While Beelzebub stoically tried to reason with him, Gabriel lost a bit of his dignity, literally begging to the young child on his knees. After Adam managed to rewrite reality in a move that turned his former father Lucifer into dust, the two of them acknowledged that it's no use trying to start another apocalypse, but telling Aziraphale and Crowley that the fight isn't over yet. They turn to each other in agreement and disappear after a firm and formal handshake; their only direct interaction throughout the entire first season.

Just as Aziraphale and Crowley think that everything is already dealt with, the forces of Heaven and Hell ambush them to execute them for their "treachery". However, they were a few steps ahead and manage to escape unscathed, leaving both Heaven and Hell utterly terrified of them in the process. As they finally get to rest for good, Crowley suggests that the next apocalypse will be "us vs. them" = "Humanity vs. the Forces of Heaven and Hell combined", indicating that Gabriel and Beelzebub will work more closely together in the future.

Fanon
Ineffable Bureaucracy is bizarrely popular, being the third most popular ship in the "Good Omens (TV)" tag on AO3, right after Ineffable Husbands and the platonic variant of that. This is despite their lack of interaction in the series proper, with that one scene really being the only time they interacted at all. People largely saw some "history" in their interactions, with many commenting that they look like a "divorced couple", especially in their interactions with Adam.

Speaking of Ineffable Husbands, Ineffable Bureaucracy most likely owes most of its popularity from the fact that it's a nice side-ship for the fandom's OTP, mostly because it's yet another Angel/Demon ship, both of whom have a connection to the main characters (though Beelzebub admittedly less so, Hastur would be more of a Gabriel-equivalent for Crowley). When it comes to depicting the main duo's reactions to the ship, Aziraphale largely doesn't mind it, sometimes even secretly shipping it himself. Crowley is utterly horrified at the idea.

The ship is so prevalent, it actually started creeping into the "Good Omens - Neil Gaiman & Terry Pratchett" tag, being the fourth most popular ship for that tag. This is despite Beelzebub only having a very minor role in the book and Gabriel only being mentioned once; they interact even less in the book than the one scene they got in the show's first season. This is most likely because of fanfiction writers tagging their fics for both the book and the show, being unaware or simply ignorant of the differences between them.

Some issues about the AO3 tagging system have cropped up for the ship, largely because of Beelzebub. Since Beelzebub is male(-presenting) in the book (only ever being referred to as the "Prince" of Hell and using a traditionally male name), yet is played by the actress Anna Maxwell Martin, people are divided on whether the ship counts as het or slash (even this wiki mistakenly tagged it as "het" at first). Official word by Neil Gaiman is that all angels and demons are non-binary, Beelzebub especially, since ze is one of the few characters using neopronouns ("ze/zim/zir", accent on the "z", like the buzzing of a fly).

Trivia

 * In the original book, Gabriel was barely a character, though he is mentioned as Aziraphale's superior. His role in the climax is instead filled by Metatron.

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