Fan Works / Friendly Rivalry

Avantasia

Scarecrow and Elderane have this in the Avantasia Protag AU series. They tease each other and try to one up each other over silly things, but are friends who won't hesitate to rescue the other if danger appears or work together when the situation calls for it.

Foxfire: Li bonds with Jet over their shared love of swordfighting, with Jet offering Li a place where they can safely practice.

Crossover

The Lion, The Witch, and The Fairy's Tail: In All Aboard the Dawn Treader, while joining Natsu, Lucy, and Happy to their next journey to Narnia, Romeo Conbolt develops one with swashbuckler Reepicheep as they sail on the Dawn Treader.

Cadets Kate Kane and Sophie Moore are girlfriends in Well-Matched, but have zero problem beating the tar out of each other in a championship boxing match. They're good sports about it afterward, and still just as much in love. Cadet Scrap, which Well-Matched is a prequel to, has a similar premise, with the same characters duking it out in a private Lighthearted Rematch and remaining on good terms.

The relationship between Varric and Dorian in Skyhold Academy Yearbook might be best described this way. They snark at and denigrate each other periodically, and play ridiculous over-the-top pranks on one another — but they also look out for, help, confide in, and respect each other a great deal. Eventually, they jokingly refer to one another as "the brother I never wanted."

Crimson and Noire: Kagami and Adrien's dynamic in regard to both their fencing and their feelings for Marinette. In chapter 38 when Kagami asks Marinette to the school dance, Adrien gives his support because Kagami is one of his closest friends and he wants to respect Marinette's decision. Likewise, Kagami checks up to make sure if he is alright with the decision and advises him to be more proactive with his feelings next time, which he takes to heart when he asks Luka to the school dance.

#14 is set in a world where Izuku was Driven to Suicide. At one point, Katsuki mentions having dreamed of a world where both of them attended U.A. together, becoming a pair of friendly rivals.
In Breathe In, Breathe Out, this is Invoked by Lady Nagant and Vlad King, who wish to encourage such relationships to form between Classes 1-A and 1-B by having them interact and compete often.

Advice and Trust has a romantic variant. After they get together, Asuka finds the idea of Shinji catching up with her Synch test scores to be a genuine thrill as it constantly pushes her to improve. Shinji has similar feelings about it, though not to the same extreme due to lacking Asuka's lust for battle.

Ffreire's NaruFox AU:
- While Ino and Sakura were both interested in Sasuke, their rivalry for his affections is much more relaxed. All the more so after Team Seven was formed and Sakura got to know him better, growing comfortable enough to tease him when he compliments Ino.
- Rock Lee wishes to follow in his teacher's footsteps by fostering a friendly rivalry with all three of Kakashi's students. When he steps in to assist Sakura against the Sound Trio, he asks if she'd be willing to take him on in a match sometime; though confused, she agrees.
Contrasting with their canon counterparts, Naruto and Sasuke are closer to this in Son of the Sannin. While they remain highly competitive with each other, the two respect each other's skills much more and their canon animosity is virtually non-existent. It also helps that Sasuke has Itachi to help curb down his ego, so when Naruto finally catches up to him and defeats him in the Chunin Exams, he's able to cope better with it.

In Prisoner of Glory, Yuri Hikawa and Kasumi Yoshizawa were this, until the latter was abruptly and tragically killed in a car accident.

Due to his more serious mindset in Challenger, Ash develops a fairly friendly rivalry with Paul. They still trash talk each other and battle any chance they get, but it's also clear they respect each other as well.
In Jessica, Cameron and his friend, "P," play Pokémon Black and White simultaneously to see who finishes first. Cameron leaves P in the dust.
In Pokémon Reset Bloodlines and its sidestories, this is shown with several characters. The most prominent in the main story is Ash with A.J., who develop a degree of respect for one another after Ash breaks A.J.'s winning streak. Since then, A.J. always awaits eagerly their next encounter, and a sidestory reveals that he considers Ash a friend.

In Device of Life, Miku and Teto considering themselves best friends, despite them being the leaders of two rivalling groups (Vocaloids and UTAUloids, respectively.)

Friendly Rivalry / Fan Works