On the Don Bluth trope page, we mentioned that because he does not own the rights to the films he directed, his films tend to suffer from Sequelitis. Bluth had absolutely nothing to do with any of the sequels to any of his movies... except this one. This is the only sequel he directed. (If you're wondering, if it isn't "better" than most sequels spawned off Bluth films, then at least it is one hell of a lot weirder.)
And, funnily enough, it barely counts as a prequel to Anastasia. It's really more of a Spinoff. The only things connecting Bartok the Magnificent to the previous film are the Russian setting and Bartok himself.
In this movie, Bartok is a traveling entertainer and a conman traveling Russia with a bear named Zozi, pretending to be a great hero and adventurer. His facade as a hero backfires when Tsarviech Ivan Romanov disappears, kidnapped by the mysterious Baba Yaga. Ludimilla, Ivan's advisor hires Bartok to get him back, and the bat finds he can't say no. So the pair set off, and Bartok might become a hero yet.
Anachronism Stew: Oh sweet sanity, where do we start? It mixes Imperial Russia (18th-19th centuries) with Russian folklore from the first millennium AD, features a fictional Romanov prince based on a fairy tale character from said folklore, and it only gets worse from there.
Applied Phlebotinum: The stuff Baba Yaga puts in the potion. Especially that mysterious glittery liquid she wrings out of Piloff.
Bad Bad Acting: Ludmilla upon "discovering" Prince Ivan has gone missing.
Call Back / Foreshadowing: Everything Bartok does during the climax is a reference to something he claimed he had done in his show at the beginning of the movie, from defeating a dragon to dousing a city in flames.
Card-Carrying Villain: Ludmilla compares herself to Attila the Hun when singing about what kind of ruler she'll be.
The Cameo: Rasputin...possibly. When Bartok returns from Baba Yaga with the potion and begins talking about its effects, an old man who bears a striking resemblance to Rasputin (albeit less evil-looking) comes over to him. Considering Bartok The Magnificent is a prequel to Anastasia, perhaps this is how Rasputin and Bartok first met?
The Mind Is a Plaything of the Body: Sort of happens to Ludmilla at the end. Baba Yagas potion makes anyone who drinks it "ten times the person" they are inside. Ludmilla drinks it thinking it will turn her into what she thinks she is: beautiful, sweet and graceful as a flower, but since inside she's rotten and villainous she instead turns into a monstrous dragon. Her manner of speaking noticeably becomes much less sweet and controlled — and more manic and aggressive - over the course of her Villain Song, but she doesn't even realize anything's changing until she gets a look at herself in the mirror. Up until that point she was still talking about how she's going to be the golden ruler to the people of Russia - by the time Bartok gets to the city she's a rampaging beast setting everything on fire, and she doesn't talk at all or act like anything but a feral monster for the rest of her appearances. Since she's the villain anyway it's less noticeable - and it's more of a "mind is a plaything of the body which is a plaything of the inner self" type deal - but it's definitely an example.
If you interpret the events to mean that Baba Yaga knew Ludmilla would steal the potion and therefore she made it a "turn into dragon" poison from the outset, then it's a completely straight version of Mind Is A Plaything of The Body.
My God, What Have I Done?: Again, sort of. Ludmilla's last words are a shocked "oh, my goodness" and gasp when she realizes that Baba Yaga's potion is turning her into a monster, but by then it's far too late.
Noodle People: Ludmilla, sort of. Everything above her shoulders is normal and the rest of her body is noodly, which looks...odd.
One-Winged Angel: During Ludmilla's Villain Song. She downs a potion Baba Yaga gave to Bartok (it was supposed to turn him into a heroic creature capable of saving the day) and turns into "the real Ludmilla" - a none-too-intimidating purple dragon.
Spikes of Villainy: Ludmilla has several superfluous spikes over her wardrobe, just in case you didn't realize she was the villain.
Super Serum/Psycho Serum: Baba Yaga's potion, which turns whoever uses it into their inner self, can be either depending on who drinks it. If taken by a repressed hero like Bartok, it would have supposedly made him into a heroic superbat. However, when taken by Ludmilla - a repressed murderous sadist - it instead turns her into a violent fire-breathing dragon. This noticeably surprises her, but not Bartok or the audience.
Swiss Army Tears: The final ingredient Baba Yaga wanted for her potion is Bartok's tears, shed because of "compassion". Awww...