'fraternal rivalry' is a correct and usable phrase in written English.
You can use it to refer to competition between brothers/siblings, usually in a playful or joking way. For example: The two brothers were always engaging in fraternal rivalry to determine who was better at sports.
At the end of which the summer now seems set fair for a sustained and no doubt deeply agonising restaging of cricket's most waspishly familiar fraternal rivalry.
Ever since Cain and Abel, fraternal rivalry has haunted us.
Always overshadowing the fraternal rivalry were the deaths of two other sons.
Like its predecessor, this "Lion King" features fraternal rivalry, family betrayal and a reluctant young heir to the throne.
The bond between two brothers becomes irretrievably twisted in "Lovers of Hate," an unusually perceptive examination of fraternal rivalry and failed ambition.
David Miliband quickly bowed out of frontline politics, saying he wanted to end the "soap opera" the fraternal rivalry had become.
The story of a young man in the Midwest on whom fortune regularly smiles even as it abandons everyone around him, "The Man Who Had All the Luck" is a play that creates the fraternal rivalries and filial conflicts that Mr. Miller revisited and honed, most famously in the Loman family.
Satan (a perfectly smug and resentful Stephen Stout), God's brother, drops in to visit, and this heavenly household is filled with laughter and bickering, sibling rivalry and fraternal bear hugs.
Behind the scenes, The Rivalry can occasionally seem more fraternal than tribal.
In the soap opera genre, brothers with a standing rivalry as well as a strong fraternal bond are considered to be a rarity.
Then came a fraternal split.
