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Sith Warrior:

Cyborg Veteran Unable to resist the call to arms, you lied about your age to join the military and were nearly killed in battle as a teenager. Under the knife your Force powers manifested. The Empire rebuilt you and trained you to be Sith.

Child of the Emperor Strong in the Force from birth, you were taken from your family at infancy and raised by the Empire. You have always been marked for greatness.

Sith Pureblood You were born with the blood of the ancient Sith flowing through your veins. Your red skin marks you as superior to the ordinary humans around you.

The only Sith backstory that seems to be specifically referenced in conversations is Sith Pureblood, which is referenced in 3 class-specific quests. There doesn't seem to be a Zabrak-specific option here.

Sith Inquisitor:

Upstart Slave Your parents were slaves and so were you-until you shocked everyone by demonstrating a talent for the Force. Now you enter a dangerous new life as a Sith.

Zabrak Criminal Arrested for killing a noble, you claimed it was justified. The Empire put off your execution when you displayed Force sensitivity.

Twi'lek Renegade Enslaved by the Empire, you murdered a guard and tried to escape. Imperials were about to execute you when a Sith realized you could use the Force.

Rattataki Acolyte An open-minded Sith Lord came to your home planet and trained your people in the Force. When the Sith Lord was killed, you and the other acolytes were thrown in the slave pens.

Twi'lek and Zabrak Inquisitors are specifically referenced in two conversations, the first of which is definitely backstory-related:

There's no Sith Pureblood-specific option, even though IIRC there's a loading screen text specific to a Sith Pureblood Inquisitor.

Bounty Hunter:

Outlaw You never knew your parents. Raised in the streets, you learned your skills through gang warfare. It's all about survival.

Cyborg Merc You fought in wars across the Outer Rim, but the rules and regulations of a soldier's life never agreed with you. You've spent years upgrading your gear and yourself.

Rattataki Gladiator Taken as a prisoner when your colony was conquered by Hutts, you survived the arena to escape and become your own boss.

Zabrak Tracker You come from a long line of manhunters and you were eager to do your family proud and join the family business.

There's one reference to the Rattataki Gladiator background on Belsavis. This is the only place where it's referenced by background (qst.location.belsavis.world.imperial.gearingup.temp_background_rattataki) instead of species (qst.utility.misc.is_bg_bh_rattataki).

Imperial Agent:

Cybernetic Experiment Eager to serve the Empire, you volunteered for a series of dangerous experiments that gave you skills equal to any trained spy in the galaxy.

Military Officer You were a battle strategist and legendary shot for the Imperial Navy until you discovered your true calling as an agent of Imperial Intelligence.

Chiss Secret Police On your conquered homeworld, you quashed rebellions, fought terrorists and maintained order for the Empire. Now, Imperial Intelligence wants you.

Zabrak Assassin As a young alien child on an Imperial colony, you already showed great promise when it came to violence. Imperial Intelligence took you in and taught you to think, talk and shoot like a professional.

Chiss is obviously referenced a lot; it looks like it's in about twenty class-specific conversations. This line from Ardun Kothe seems to reference the Zabrak Assassin background, but triggers for Rattataki instead?

There's no Rattataki-specific background listed.

Jedi Knight:

Force Sensitive Orphan Left on the Jedi Temple's doorstep as a newborn, you were raised entirely by Jedi and know little of the greater galaxy. Your true heritage is unknown.

Zabrak Swordsman Brought to the Jedi Temple as a child, you have always found your calm and connection to the Force through the intricate saber routines taught by your masters.

Miralukan Survivor The Sith devastated your world, forcing you to flee. Thanks to your powerful Force sensitivity, you found quick refuge among the Jedi.

The only dialogue reference I could find is when fighting Lord Nefarid, a Miraluka Knight will say they don't need to see him to fight him. There aren't Twi'lek or Mirialan-specific options despite both being available by default. There are lines from Kira that were probably supposed to reference these backgrounds, but for some reason it's assigned to trigger for Chiss and... also Chiss?

Jedi Consular:

Jedi Scholar Fascinated with the history and traditions of the Jedi, it was your research of the Force that unlocked your amazing potential.

Twi'lek Visionary You were born in space and grew up in a small colony of ex-slaves. When you started seeing visions of things to come, one of the wise elders took you to find the Jedi.

Mirialan Wanderer Raised in a relatively primitive tribal society on Mirial, unexplainable visions drove you to roam the galaxy until you found the Jedi and your true calling.

Miralukan Prodigy When the Jedi found you abandoned as a child, you were already performing feats one would expect from a Jedi who had been training for years.

The only dialogue reference I could find was that a human Consular tells Nadia's father that they were a prodigy as well. There's also three lines from Syo and from Yuon that can play, but it looks like only the first ever will as they have the same conditions. I assume 1 was originally the Miralukan Prodigy, 2 was the Jedi Scholar, and 3 was the Twi'lek Visionary and the Mirialan Wanderer.

Smuggler:

Corellian Nomad Descended from people of fierce independence and pride, you're always on the lookout for adventure and any job that'll turn a quick credit.

Outer Rim Pirate You're utterly ruthless and only in it for the money. If that means becoming more machine than human to gain an advantage, that's what you're going to do.

Twi'lek Rogue You were smuggling your own people out of slavery, but the Empire's aggressive expansion forced you to take other jobs to survive.

Mirialan Trader You've tried your best to keep your business legitimate, defend your cargo and protect your honor. Not always easy stuck between the Imperials, the Republic and the Hutts…

Twi'lek Smugglers are mentioned in one line, but it's not backstory-related.

There's no Zabrak-specific backstory, once again.

Trooper:

Republic Elite You were groomed from youth to lead soldiers into battle. You've had the best teachers and trainers in the entire Republic, and it shows.

Outer Rim Veteran Blooded in the terrible conflicts of the Outer Rim territories, you gave the Republic permission to put you back together and you went back out for more.

Zabrak Guerilla Fighter The Empire invaded your world, and you fought back valiantly until Republic reinforcements arrived. Now you continue the fight on other worlds.

Mirialan Youth Leader You were active and well loved in your community from a young age but secretly disappointed in your pacifist society. When the Republic recruiter came calling, you shocked everyone by signing up.

The first three do appear in dialogue with Jorgan, and the Zabrak Guerilla Fighter is referenced again on I think this is Corellia? Mirialan Youth Leader isn't referenced anywhere.

I Feel Like This Section Also Needs A Header

Interestingly, there are no Mirialan-specific lines in the entire game. Miraluka has one, Rattataki has a couple, and Zabrak has two each for Inquisitor and Trooper. Sith Pureblood has a few in the Sith Warrior storyline. Chiss definitely has the most.

This obviously isn't a comprehensive list of dialogue options. The Consular lines, for example, I stumbled upon completely by accident, so it's safe to assume there are more scattered around-- there's just not a good way to find them without manually trawling every conversation in the game. (Which I'm not doing.)