Vegito was supposed to be permanent. Once a pair of individuals put on the Potara Earrings, they were meant to be eternally fused, inseparable for the rest of time. However, Vegito was put under extraordinary circumstances while fighting the demon Buu, and after being absorbed into him, Goku and Vegeta were transformed back into their original halves. At least, that's what happened in the original timeline, known as Timeline-1, from which all other branches of reality were derived. In another branch, known as Timeline-11, Vegito was never absorbed, and as a result, was never separated.
The remarkably powerful fusion of Goku and Vegito demolished the Demon Buu, saving the universe from the most dire threat it had ever known.
Despite Earth and the rest of the universe being rescued from the brink of destruction yet again, this still carried some unfortunate implications for Goku and Vegeta, who were now forced to share one body for what they could only assume would be the rest of their lives.
Goku and Vegeta returned to their home, or rather, one of the two of their homes. It was decided between them that Bulma's house was the best bet, as she would probably have the more measured reaction to their fusion.
Goku and Vegeta noted that the experience of being Vegito had changed immensely from when they were first fused. During the dawn of their fusion, they were perfectly united, moving in sync, no disagreements, only action. Now, their movements were clumsy, and split between the two. The closest comparison to the difficulty they had steering their shared body would be to two people attempting to play a video game using the same controller.
"Dammit, Kakarot, why is this so difficult than before?" Vegeta said or rather, thought, as they landed on the lawn of the Capsule Corporation, stumbling and lurching along the grass in a cruel mockery of walking.
Since Goku and Vegeta shared a mind and body, they were now longer allowed the privilege of private thoughts. Apparently, there was little difference between Vegeta's internal monologue and the things he was willing to say out loud.
"I don't know," Goku mused. "Maybe it was because we needed to fight Buu, and we had to work together."
"That's the stupidest thing I've ever heard!" Vegeta shouted, his anger echoing throughout his shared mind, followed by a much quieter but still audible: "well, not really, but it's still stupid, Kakarot" proving that, perhaps, there were some things that Vegeta used to keep to himself.
"Hey Vegeta, you know what I've always wondered?" Goku said, as Vegito continued to stumble vaguely in Bulma's direction.
"What is it, Kakarot," Vegeta said with a mental sigh.
"How do you spell my Saiyan name? Are there two ks, two rs, or both? Or maybe it's neither…"
"What do you mean you don't know how to spell your Saiyan name? You're a disgrace to your race, Kakarot! To a Saiyan, your name is the second-most important thing you are given by your father, next to your life! And you're telling me you don't know how to spell it?"
"I was just asking," Goku said. If he wasn't simply a voice inside the brain of their mutual body, he would've crossed his arms and pouted mockingly at Vegeta.
"Shut up!" Vegeta said, even though their conversation was completely silent to any outsider.
"That woman can't be too far from here." Vegeta desperately tried to think of Bulma's location, and to banish any potentially sentimental thoughts he may have had about her, and her inviting smile, or the resilience she possessed that reminded him of his own mother, or the way she continued to love and tolerate him, even during his most outrageous moments or the offspring they shared to the furthest corner of the mind that was no longer his.
Goku noticed Vegeta's ungainly attempts at mental repression, and out of respect for his longtime friend, refused to probe into the corners of his psyche that he had made forbidden. To make absolutely certain that Goku didn't reach them, he placed other thoughts and memories of his between them, namely, recollections of the atrocities he had committed and how much he enjoyed them at the time. Goku took the hint, and ignored that part of Vegeta's consciousness even more.
Fortunately for the two of them, they no longer needed to trudge around the Capsule Corp with their bizarre gait, as Bulma had found them.
"There you ar-"
Bulma paused, and quickly deduced that the stranger with an outfit that was a clear combination of Vegeta's battle armor and Goku's orange gi must've been a combination of her husband and Goku, somehow.
"Yeah, hey," Vegito said, scratching the back of his head, speaking Goku's voice with an echo of Vegeta's, or perhaps, the other way around.
"We got fused together while we were fighting Buu...permanently," he continued.
"I can see that," Bulma said, circling around their fused body, examining it carefully, letting out a soft "hmmmmm," as she identified which features belonged to her husband and which belonged to Goku. She recognized Vegeta's unmistakable firm rear almost immediately, and noted that the upper part of his body was slightly broader and more muscular than Vegeta's, like Goku's.
"You said you couldn't defuse, right?" she said, crouching down to see whose feet Vegito had gotten.
"Yes," Vegito replied. "So, are you going to find a way to separate us, or are you going to keep staring at our shoes, woman?"
Bulma huffed, rising to her feet, and giving Vegito a firm knock on the forehead.
"Yeah, Vegeta's definitely still in there."
She backed up, and crossed her arms. "The solution's obvious, I'm surprised you two haven't thought of it already. We'll just use the Dragon Balls! Two brains, and you still can't figure out what's right in front of you!"
"Right," Vegito said with a huff.
Bulma left for a few seconds, and returned with the Dragon Radar.
"Krillin and Tien could probably keep this a secret, right?"
"Yeah, why?"
"I'll have them help us find the Dragon Balls. It'll take too long if you do it by yourself, err, selves."
They nodded. Krillin knew how to keep a secret, and Tien didn't talk much to anyone besides Chiaoutzu.
"Hey Vegeta, we can use the instant transmission in this form, right?" Goku said aloud.
"Yes, Kakarot," he replied, also aloud. "Also, if you're just talking to me, keep it inside your head, otherwise it'll look like someone's talking to themselves."
"Right," Goku said, still speaking out loud, through Vegito.
Vegeta grumbled physically, while Goku continued to talk, since Bulma needed to hear what he had to say.
"Well, if we have instant transmission, that'd make finding the Dragon Balls even easier!"
"Right," Bulma said. "I'll make a Dragon Radar for all of you, it shouldn't take longer than a day to find them all."
Bulma kept her promise, giving Dragon Radars to Tien, Krillin, and Vegito. Krillin tried to keep his thoughts about what was going on to himself, but he couldn't stop glancing over at Vegito, wondering what it must've been like for his friend to be trapped in the same mind as the terrifying Saiyan Prince.
The trio, or rather, the four of them went their separate ways, collecting the Balls from around the world and bringing them back to the Capsule Corporation. In the meantime, Bulma picked up her phone, and discovered that Chi-Chi had left her no less than 55 messages.
She had concerns to put it lightly, since she hadn't seen her husband since he went to fight Buu. Bulma assured her that he was fine, and that him and Vegeta needed special medical attention after their fight that was only available at the Capsule corporation.
"You should've invited me over, then! I have a right to see my husband!" she said, beginning to lose her composure.
"You're right," Bulma said. "The medical procedures are very delicate, can you come over tomorrow evening?"
"Yes," Chi-Chi replied. "Oh, I hope he's okay."
Bulma hung up, and sighed. Tomorrow evening would be more than enough time to collect the Dragon Balls, summon Shenron, and return their husbands to normal.
By the looks of it, Krillin, Tien, and Vegito had already returned with the last Ball. Krillin and Tien dismissed themselves, promising silently to leave this weird incident behind them.
Bulma stood by, as Vegito summoned the Dragon. The balls glowed with a yellow light, the day's blue sky turned a stark black, and a great beast erupted from the pulsating orbs.
"State your wish." Shenron said.
Bulma pushed herself in front of Vegito, deciding it'd be fun to make a wish for a change. Goku and Vegeta didn't particularly care, as long as they were separated.
"Great Dragon!" Bulma said, putting some dramatic flair in her voice, clearly enjoying herself quite a lot.
"Separate this guy back into Goku and my husband," she said, pointing towards Vegito.
"I cannot grant your wish," Shenron said. "I cannot do anything exceeding the power of my Master, and he was fused by the power of the Supreme Kais."
"That's...oh," Bulma said, trying to think if there was some sort of alternative wish she could make that'd still help the situation. She got three, after all.
Before she could think of anything, her attention was diverted by Chi-Chi, who was darting angrily over the Capsule Corp lawn, kicking up grass and dust beneath her feet.