Perhaps Naruto discovers a new power or a new path of training that is not recognized by Tsunade. He feels that he can't fully explore this new aspect while staying under her supervision. He then chooses to leave in order to pursue this new discovery freely.
One reason could be a mission. Maybe he gets a top - secret mission that requires him to leave the village under Tsunade's leadership without her direct involvement. So he has to leave without much notice.
One possible reason could be to search for a powerful artifact. In many fanfictions, there are often hidden and powerful items that can change the fate of a village or even the whole ninja world. Naruto and Tsunade might leave Konoha to find such an item before their enemies do. This would be a dangerous mission, so they keep it a secret from the rest of the village to avoid panic.
One possible reason could be to search for a powerful artifact. Maybe there's a rumor about an ancient item that can save the ninja world from a new, unknown threat. Naruto, with his adventurous spirit, convinces Tsunade to leave Konoha and embark on this quest.
One possible reason could be a misunderstanding. Maybe Tsunade received some false information about Naruto that made her think he was a threat to the village, so she took actions that ended up hurting him.
Maybe Naruto discovers some dark secret about Tsunade's past that she has been trying to hide. For example, something related to her lost loved ones. Naruto, in his usual straightforward way, confronts Tsunade about it without realizing how much it would hurt her, and thus makes her cry.
It could also be related to personal relationships. Tsunade might have been hard on Naruto in a way that he felt was unjust. Perhaps she was overly critical of his unorthodox ninja techniques or his impulsive nature, and over time, this built up into hatred. Naruto, being someone who values respect and recognition, could have resented her for not seeing his potential in the way he wanted.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he has reached a level in his training where he believes he needs new experiences and challenges that Jiraiya can no longer provide.
In fanfiction, it could be due to a prophecy. Suppose there was a prophecy that Naruto would bring great danger to the village in the future. Tsunade, despite her bond with Naruto, might believe in the prophecy and banish him to prevent the foreseen disaster. Also, it could be related to Naruto's power. If his power was growing too rapidly and becoming uncontrollable, Tsunade might banish him to protect the village from any potential harm his power could cause.
One possible reason could be political pressure. Maybe some powerful clans in the village see Naruto as a threat due to his growing power and influence. They might force Tsunade's hand to banish him to maintain their own status quo in the village.
One possible reason could be that Tsunade makes a decision regarding the village that Naruto strongly disagrees with. For example, if she refuses to send help to a neighboring village in need, Naruto, with his strong sense of justice, might get angry. He's always been about protecting and helping others.
Another reason might be that Naruto is seeking a greater purpose. In the fanfiction, his 'godlike' state could make him realize that there are threats out there that he can't face while being tied to Konoha. He may leave to protect the world on a larger scale, believing that he can be more effective away from the village.