So the red part of Sombra's eyes - are those his real pupils or is that a magical effect resulting from his use of Dark Magic?

When Celestia demonstrated to Twilight what Dark Magic was in the Crystal Empire episode, we saw both her pupils and the whites of her eyes turn green. So, with that in mind, wouldn't that mean that dark magic would only turn a caster's eyes green and cause purple magical 'streaks' to leak out, instead of than turning their pupils red as well?

Because if that were the case, then Celestia's pupils should have turned red, too - which they didn't.

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It is assumed that it is indeed the effect of the dark magic on its user.

As for the Celestia thing, it could be blamed on inconsistent design on the animators, if you want the simple solution.

3360538 Perhaps it's because Sombra used or had so much dark magic within him, that the green eye thing became permanent, and his eyes being red are a side effect of using the magic for so long. Or he turned himself into a demon. That's another option.

Sombra's eyes are red from the effect of dark magic, as are his reptilian pupils. There is a theory out there that Sombra has dragon blood within him, thus his animalistic nature and eye color, but it's merely a theory at best.

As for the green corneas and the purple flames from the eyes, they are only active whenever feelings of anger are felt or any type of aggression whatsoever. When Sombra is attacked, he is shocked and thus the magic disappears from his eyes.

3360538 I've seen an example of where his eyes are red before his change, personally I don't know for sure, but that eye color isn't so farfetched in Equestria.