His relationship with Sasuke would likely be very different. Without the shared experiences in Konoha, their initial rivalry might not be as intense. Naruto might not be as desperate to bring Sasuke back. And his relationships with his other Konoha friends like Sakura and Kakashi would also change. He'd be more connected to Yugito's friends and allies instead.
One possible plot twist could be that Naruto develops unique jutsu influenced by Yugito's Nibi powers. Since Yugito is a jinchuriki, she might teach Naruto some special techniques related to chakra control and dealing with tailed beasts. For example, Naruto could learn a new form of Rasengan that incorporates some of the Nibi's chakra nature.
In this fanfiction, the story might focus on how Naruto's character develops under Yugito's influence. Yugito could introduce Naruto to a different side of the ninja culture that he may not have been exposed to otherwise. For example, she might have connections to certain clans or traditions that Naruto becomes a part of. This could lead to new friendships and alliances for Naruto, as well as new enemies who might not like this new 'Naruto' that is being shaped by Yugito. Also, Naruto might learn different stealth techniques from Yugito, as she is from a different village and has her own set of skills in that area.
Naruto's training would involve a deeper understanding of tailed beasts from an early age. Since Yugito is a jinchuriki, she could share her experiences and knowledge about the Nibi with Naruto. This would give him an edge when it comes to dealing with Kurama. He might learn how to better communicate with Kurama or even gain some new abilities related to the tailed beasts' chakra earlier than in the original story.
One possible plot could be that Naruto grows up with Yugito Nii's unique teachings. He might learn different jutsu styles from her. For example, he could master some of the Nii clan's secret techniques. Yugito Nii could also instill in Naruto a different set of values compared to what he would have learned in Konoha on his own. This could lead to Naruto having a more nuanced view of the ninja world and different relationships with the other villages.
One great 'Naruto Yugito Fanfic' is 'Yugito's New Path'. It shows an interesting alternate story for Yugito. In this fanfic, she gets to explore new powers and relationships that were not explored in the original Naruto series.
Naruto's character would likely be more calm and collected. Yugito Nii's influence might make him think before he acts. He would still be determined but in a more refined way.
Another theme is cooperation. They both are strong ninjas, and fanfics often depict them working together against powerful foes. Their combined skills and abilities make for some exciting plotlines. For instance, in a story where they face a powerful rogue ninja, their cooperation is key to victory.
I'm not sure specifically as there could be many different interpretations of such a fanfic. It might involve the characters Naruto and Yugito in some kind of unique adventure or relationship. Maybe it's a story that combines their different ninja abilities and personalities in a new and interesting way.
Yes, there are some out there. You can try searching on fanfic websites like Fanfiction.net or Archive of Our Own. Just type in 'root naruto x yugito' in their search bars and you might find some interesting stories.