You see, I'm making a Twilight x Tempest fic, but I have one, small problem plaguing me.
Should I use Tempest's real name in it, or should I not? It takes place right after the movie.

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In the story I’d say yes, because I’d imagine now she would try and go back to who she was. In the descriptions I’d say no because someone quickly glancing over it might have it spoiled.

So, I've been thinking about this as well... After reading a couple of fics on the subject of after movie relations, it seems that most people have taken to calling her Tempest still. (Because... well, it's cooler.) I honestly think that when I get around to writing mine it will be using Tempest, and not... Ugh Fizzlepop Berrytwist. But honestly, since the character is so new, I doubt anyone is going to hound you for using either name since there's not a popular way to refer to the character quite yet.

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I would say no because I think tempest suits her better.

I feel like if she wanted to go by Tempest she would not have said her real name. It is of my opinion that she wouldn't want her name to reference her past either. Fizzlepop Berrytwist all the way, but I would just write her as Fizzle for short.

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It's up to you as a writer to show what she thinks. If she's ashamed of her stint as the Storm King's right hand mare then maybe she wants ponies to call her Fizzlepop. Maybe the pain of her childhood is still too much so she wants to stick with Tempest until she can process her emotions. Maybe she embraces the anti-hero role or wants to be intimidating for some reason like maybe now she's the commander of Celestia's airfleet so she stays with Tempest. Then again, you could get some humor out of something like "Commander Fizzlepop the Dread."

The real mystery is what her cutie mark should be. I see some fans using a bowl of ice cream from a similarly named G3 pony, others make up their own, like firework explosions or a tornado.

I have kept to calling her Tempest but reference her reaal name every now and again

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How about just making the naming issue part of the story. her struggle to figure out who she is now she has friends. I also think it'll depend on the character.

Twilight will say tempest, then realise her mistake and say her real name instead along with an apology.
Fluttershy will use Tempest mostly, but will try to use her real name. She'll be nervous about it though. Unsure which is best, and partially scared of her upsetting Tempest enough to make her go back to being...evil.
Rarity, will use Tempest and reason it's a fashionable name and if she chose it, then she shouldn't have shame for having done so, regardless of her past circumstances.
Rainbow Dash, will use Tempest because it sound cooler...like 20% cooler.
Pinkie Pie will INSIST on using Fizzlepop Berrytwist because...well you saw her face at hearing it.
Applejack, will ask and go with what suits Tempest most.
Most ponies that meet her will call her Tempest. It'll be down to Tempest or others (like Twilight) to correct them.
Discord, mixes and matches the names in many a way in his typical chaotic fashion.

I think if you write it with this sub-plot (if the sub-plot works for you in the fic that is) then it'll give you time to decide which name she'd wish to go by. Personally I think she'd settle for Tempest Shadow, but her friends would get to know her real name. Like Tempest Shadow is the name she'd go by in the Equestrian Royal Guard, if she joined it.

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Maybe as a nickname for special ponies.
In a romance it was used so far, that Twilight always called her that way to make her listen, so she knew that it was serious for her.