Since so many people don't know how easy it is to miss Brionac, I hope this step-by-step guide helps.
Brionac - Judith's strongest weapon (Unless you use her Fell arms weapon)
1. After finishing Shaikos Ruins, you will be told to return to Aspio, specifically Rita's house. Go there and view the scene. Finish that scene and leave. You will get another scene with Rita joining you, then after that, go immediately back to Rita's house and check the blastia on the left. If you don't know what I mean, start clicking A on the left wall and you'll get it eventually. Rita will then name the blastia Victoria after the scene.
2. After finishing Ehmeed Hill, go IMMEDIATELY back from the way you exited and then go back to the blastia. Rita gives the blastia the name, Ekaterine(?).
3. After Brave Vesperia is created, go to Capua Torim and speak to a woman by a tree. It's on the upper level. A scene occurs where Judith wonders off and attacks a nearby blastia while the others speak with the woman.
4. In Nordopolica (the first time you get there), talk a man by the pot. Judith should once more leave the group here and attack a nearby blastia.
5. After Judtith comes back @ Mount Temza, go to Capua Torim and speak with the same woman as you did in #3 and here, Judith admits that she attacked the blastia from way back when. (You should be able to fly with Ba'ul now. You can finish the next few steps immediately after this one.)
6. Go to Ehmeed Hill after finishing #5 from the left. And yes, I said LEFT. Not right. Here, you see Rita talking to Judith about the incident where Judith destroyed the blastia.
7. Head to Ghasforost after finishing #6, then go to where you first controlled both Yuri and Judith. It's the right room >>. In there, you will get a scene. After this, go to Capua Torim and speak to Kaufman at the Fortune's Market HQ. A scene occurs where Judith asks Kaufman about Brionac. Then go to Dahngrest and talk to the merchant at the Fortune's market store there (the store with the sign, #1 not store #2).
8. After Aurnion shows up, go to Zaphias and a scene should occur where Judith asks Yuri to visit the Lower quarter. Enter Zaphias through the Lower quarter and try to go out of there through the north exit. A scene starts here where the girls (Rita, Estelle, and Judith) say they want to destroy the blastia in the castle. Go to the Noble's quarter and try to enter the castle, A scene starts here and follow through with it. The scene is where the girls destroy a blastia in the knight's dining room.
9. Rest @ Aurnion's inn. A scene with the 4 spirits occur and after finishing that, you will wake up. Get out of Aurnion and re-enter for part 10.
10. After re-entering Aurnion, rest at the inn once more to get a scene in the morning where Judith and the others have a conversation and Judith starts to think about giving up her search on the Brionac. Leave and re-enter Aurnion one more time.
11. Change the show-up character to Rita, and check the blastia in the middle of the town. (the "rock" you see as soon as you enter Aurnion). A scene occurs where Rita talks about the blastia and whatnot. I forgot what they said. But Rita and Judith are both in that scene.
12. Go to Caer Bocram (the city where you thought Gusios first) and then go to the first house on the right. A scene occurs with the Blood Alliance guild remnants and they fork over the Brionac after being threatened by Rita and Yuri. After all this, you finally get Judith's strongest spear.
Thanks for this Vesperiahelper, I followed Kouli's, but it can be a bit hard to do them in order as it isn't as organized as I like (but instrumental in helping with sidequests of course) and I tend to get lost with this sidequest.
You cant finish the whole quest because although you can do #'s 3-11 without parts 1 and 2, you need those two parts to get Brionac on part 12. So if you didn't do them, you need to do another walkthrough to get Brionac