Tamaki and Fat Gum have a teacher-student relationship and whenever Tamaki works as a hero, he can be seen talking or fighting alongside Fat Gum, showing that he is always working at his agency. Although their personalities are opposite, they show great teamwork and complement each other.
Tamaki seems to perceive his working relationship with Fat Gum a little more negatively due to his negative personality, but despite those thoughts in his mind, Tamaki never left his agency and does not seem uncomfortable fighting side-by-side with Fat Gum, except when he talks about his fragile mentality or when he compliments him during or after a fight.
Fat Gum thinks highly of Tamaki and sees great potential in him, he is also seen constantly praising his abilities for being above most professional heroes and complimenting him on his attacks and maneuvers. Fat Gum is also direct and honest with Tamaki when he has to be and realizes Tamaki's flaws and weaknesses and what he has to improve on, such as his shy personality. But Fat Gum is shown to care deeply for Tamaki when he is in danger or when he is put under a lot of pressure, behaving like a father or older brother to Tamaki and seeing the efforts he had to go through to perfect his Quirk.
Although I know it is still unknown how they met, if he knows that at some point Fat Gum hired Tamaki to work in his agency and one day Tamaki recommended Eijiro Kirishima to work in his agency, surprising Fat Gum, but quickly accepting Eijiro at his agency, showing that he trusts Tamaki's judgment.
Fat Gum is seen for the first time in history alongside Tamaki and Eijiro while on patrol in a nearby city. After Fat Gum finishes talking to Eijiro, he tells Tamaki that he needs to fix his terrible mentality as he has a lot of potential, which emphasizes Tamaki a bit. Tamaki says that he thinks the only reason Fat Gum chose him as an intern was to torment him and calls it workplace bullying, but Eijiro assures Tamaki that Fat Gum was only trying to encourage him.
Before they could continue their conversation, a group of criminals appear and Fat Gum jumps in to catch them. Fat Gum catches them with his fat, but one of them escapes and is quickly caught by Tamaki's tentacles. When the criminals are handed over to the police, Eijiro congratulates Tamaki and Fat Gum tells him that Tamaki's Quirk has already surpassed many professional heroes and that the only thing he should focus on changing is his fragile mentality, while Tamaki was paralyzed by the crowd of civilians who congratulated them.
FaTama is mostly loved inside and outside of Japan and is sent despite being a teacher and student relationship. Unlike other Ships like EraserDeku, some fans have no problem shipping them because their ages and height don't differ much.
Mostly those who send them like their opposite personalities and compare them to other Tamaki Ships and are amused by their funny and mocking dynamic on the Fat Gum side, but what they like the most is the fact that despite their differences they get along good and the more moments you have the more they start to send them.
- Fat Gum seems to be or considers close to Tamaki, as he is the second person to refer to him by his first name, something that only Mirio did.
- Because Fatgum is a man who needs to eat a lot, he helps to supplement the requirements of Tamaki's Quirk that he needs have food in his stomach.