Yes, there are quite a few. You can start by looking on fanfiction platforms like Archive of Our Own or FanFiction.net. Just search for 'naruto x narumi x satsuki' and you'll likely find some interesting stories.
Common themes can include self - discovery. The three characters may be on a journey to understand themselves better while being in each other's company. Power - up or training arcs can also be a theme, as they work together to become stronger. Additionally, there could be themes related to family, like how they create a sort of surrogate family unit among themselves, especially if they are all orphans or have difficult family backgrounds in the Naruto lore.
One common theme is friendship. They all grew up in the same village and have different relationships with each other. Fanfictions might explore how they support each other through tough times. For example, during a difficult mission.
It could be a rogue ninja from a forgotten clan who wants to steal the unique powers of Naruto, Naruko, and Satsuki. He sees their combined potential as a threat to his own evil plans of taking over the ninja world. This antagonist might have some ancient and powerful jutsu that the three heroes have to figure out how to counter.
A more complex fanfiction could revolve around a shared goal between Naruto and Satsuki. For example, they both discover that there is an ancient evil that has been influencing events in both their universes for a long time. They have to put aside their differences (cultural and otherwise) and work together. Along the way, they might have some disagreements but ultimately grow to respect each other more as they fight side by side.
You can try some popular fanfiction websites like FanFiction.net or Archive of Our Own (AO3). These platforms usually have a wide variety of fanfictions, including those related to Naruto and Satsuki.
One popular story might be 'The Unseen Bond' which delves into the complex relationship between Naruto and Satsuki in a unique way. It shows how their different personalities and goals can bring them together in unexpected situations.
Maybe a fanfiction where they are all in a different timeline. Naruto and Naruko are from a more peaceful Konoha, and Satsuki has been sent there as an ambassador of sorts from her village. At first, there are misunderstandings, but as they get to know each other better, they start to work together to improve relations between their villages. They learn from each other's cultures and ninjutsu techniques, creating a unique bond.
Naruto's sunny and determined personality would contrast with Satsuki's more reserved and calculating nature. Yakumo, with her somewhat introverted and powerful self, would be in the middle. Naruto could bring out the more adventurous side in Satsuki, while Satsuki might help Naruto think more strategically. Yakumo could use her genjutsu insights to understand both of them better, and they would have to learn to respect each other's differences.
One plot idea could be that Naruto, Yakumo, and Satsuki are on a mission together in a strange, new land. They have to rely on each other's unique skills. Naruto's strength and determination, Yakumo's powerful genjutsu, and Satsuki's intelligence all play a part in their survival. They start off as reluctant teammates but grow closer as they face various challenges.