A/N: this was one of two what if I had in mind (the second one being Gogeta vs Merged Zamasu). But, what if Goku had a pair of Potara earring with him just in case they would be in any similar situations like in the future. What if they were forced to use those and become Vegito against Broly?

Now, we need to look at the personality between the two when in battle.

Vegito exhibited Vegeta's cockiness and inclination for taunting others, and enjoys toying with enemies, but was not as serious, having inherited Goku's easygoing, cheerful nature. In Super, it was mostly the same except he was far more serious than last time due to his opponent not needing to worry about any loved ones' safety and now knowing about his fusion's time limit. But, he was still somewhat cocky, tending to mock his opponent for things like sloppy fighting and ignorance of the situation. 

While Gogeta had that similar to him, he was more practical and efficient than his Potara counterpart. Whereas Vegito typically enjoys belittling his opponents, Gogeta rarely ever spoke during battle save for brief taunts, and he tended not to fool around with his enemies once he recognized them as a threat.

So with that, I may have a thought in mind how it would go. Given that Vegito is much more cocky, he would likely taunt broly through some times in the battle and would mock him for how he would fight, burning through some of the minutes the fusion has. I think he would get some major hits on Broly and maybe have broly in the state he was in during the gogeta fight. Though, Vegito will probably run out of time before he could try to deliver any final blows.

But, that's only me. If anyone else has an idea on how it would go from the gogeta fight with Broly, just leave it in the comments.