The final saga of Dragon Ball Z saw Goku and Vegeta fusing together to form Vegito in order to defeat the evil Majin Buu. This form combined both of their personalities and features, creating a Saiyan warrior who was the best of both Z Fighters. One element that wasn't amalgamated, however, was the mortality of one of Vegito's halves.
At the time of Vegito's first fusion, Vegeta was dead, but this isn't reflected by way of Vegito having a half halo or any other sort of signifier. There's never been an explanation for this, but there are some possible ways to figure out why the wearing of the Potara Earrings combined a dead and living fighter into a wholly alive one.
Potara Earrings
The Potara Earrings worn to fuse Goku and Vegeta had been introduced beforehand as mere decoration for the auditory sensors of the Kais. However, eventually, Akira Toriyama established that they could also be used to fuse individuals into powerful new forms. Nonetheless, these are only meant to be worn and used by Kais, meaning that fusions such as Vegito are temporary, typically lasting for only an hour before splitting back apart.
At the time, Vegeta was dead, having sacrificed himself in an effort to defeat Majin Buu. King Yenma was merciful to the prideful but just Saiyan royalty, allowing him to return to fight Majin Buu again but keeping him dead. Despite being deceased, the fusion of Goku and Vegeta that forms to fight Super Buu is completely alive.
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Why Vegito Didn't Have a Halo
It's explained in the episode "Union of Rivals" that the fusion of a dead and a living person will always be living, at least in the case of a Potara Earring fusion. After all, Gogeta had a halo due to a deceased Goku and Vegeta combining via the Fusion Dance. There aren't any other examples of a dead and living person fusing, however, so the unique Vegito situation could be explained by the special stipulation that King Yenma afforded Vegeta.
There's also the fact that Vegito is treated as an entirely separate character from his two "ingredients." Unlike Gogeta, Vegito is considered a separate entity and not merely a combined form of Goku and Vegeta. Rather, he's a new person brought to life by the fusion of the two Saiyans. This could explain why he lacks Vegeta's halo.
As mentioned, much of this is circumstantial given the uniqueness of the situation surrounding Vegito's creation. This is complicated further by neither Goku nor Vegeta being Kais, making their fusion via Potara less "pure." Whether or not another example of an earring fusion in a similar situation will be used in the franchise remains to be seen.