Zeetha, Daughter of Chump

Zeetha, lost princess of the Lost City of Skifander, is Agatha's traveling companion and has the unique relationship to her of "Kolee-dok-Zumil." She's easy to pick out in a crowd, with her waist-length green hair, two unusual swords, and an occasional glimpse of what some readers call "the navel of death," not without reason.

Backstory
We know more of Zeetha's backstory than we do of most of the Girl Genius characters, even important ones, because of the long narrative spun by Madame Olga after Zeetha went berserk at camp one day. Zeetha is a daughter of the ruling house of Skifander, which was "discovered" by some expedition a few years back. The Skifandrian Powers That Be decided to send one of their own back with the expedition to see what the rest of the world was up to, and picked Zeetha for the job. Unfortunately, this didn't go well, as she got sick and feverish on the trip, and just as she was getting over the fever, the expedition's airship was attacked by pirates. (It's speculated, but not proven, that this pirate band was under the command of none other than the infamous Bangladesh DuPree.) The pirates killed all of the expeditioners except Zeetha, whom they were planning to sell -- apparently they found her hair exotic -- and therefore treated her relatively well while holding her captive. BIG mistake. Once she was back to sword-swinging good health, she got out of captivity, killed all of her captors, shot down the other ships in the pirate fleet, and moved on to wipe out their home fortress. This latter, however, was a mistake, since in doing so she also killed everyone who knew where she'd been picked up. As a result she's now a "lost princess," with no idea how to get home or exactly where "home" is. The next few years were spent searching for some way home - with no luck, since hardly anyone had ever heard of Skifander - and traveling with Master Payne's Circus of Adventure, where she's performed various martial-arts tricks. When she discovered that Agatha had heard of Skifander -- unlike anyone else she'd ever met to that point -- yet didn't know enough about it to help get her home, that was just too much for the poor, sweet thing (not...), and she chopped up the camp.

Eventually she apologized and resumed searching for a way back to Skifander. At this time, though, protecting her  zumil Agatha is a little more urgent.

Kolee-dok-Zumil
The Kolee-dok-Zumil relationship she enjoys with Agatha is hard to describe concisely, so go read the article. Zeetha adopts Agatha into this relationship after chewing her out over her complacency, when it looks like the ruse to convince Baron Klaus Wulfenbach that she's dead has succeeded. As Zeetha points out, there's one serious flaw in Agatha's plan: she's not really dead. Klaus will eventually figure out that he's been duped, and he's not going to like it.

Kolee-dok-Zumil is only available to one person other than the Kolee's own daughters (or so Zeetha says), so why does she choose Agatha for this role? You'd have to ask Zeetha to be sure, but:
 * Agatha, being a Spark, has a certain charisma;
 * It's very much in Zeetha's own interest to keep Agatha alive, so that she can make introductions to Uncle Barry and help Zeetha discover a way back to Skifander;
 * Zeetha sort of owes Agatha one, for giving her hope by convincing her that Skifander really does exist and wasn't just a figment of her feverish imagination;
 * Zeetha, who is tough as nails, likes having a nice, soft novice around to whip into shape, and Agatha fits the bill perfectly;
 * And of course, above all, Zeetha is one of the Good Guys, and therefore fated to help Agatha out.

Anyway, the relationship works pretty well. Zeetha is the first person other than Punch and Judy (to the extent that they are "persons") who really exerts herself to keep Agatha alive and well. (Well, there was Lars, who exerted himself to a terminal degree on the same goal, but he's no longer around.) The two are presently separated while Agatha tries to fix Castle Heterodyne, but a reunion is coming, probably.

Relationship to House Wulfenbach
Needless to say, Zeetha is right in the middle of the action when the Baron comes calling at Master Payne's Circus of Adventure. Klaus' loud "YOU!!" suggests that he knows who she is, although the next moment reveals that Zeetha is unaware of any connection. At first she simply converses with him in Skifandrian, but when Klaus decides (for reasons Zeetha cannot possibly know) to kill Agatha, she does what a good Kolee does to protect her Zumil: she tries to kill him first. It is reasonable to say that the relationship goes downhill from there.

As far as we can tell, Zeetha has never met Gilgamesh Wulfenbach. However, Klaus thinks that there are negative vibes between them somehow -- about as negative as you can get, as in, Zeetha has been sent to kill Gil. We know that isn't true -- Zeetha's wandering into this whole scene is a fluke -- but Klaus thinks otherwise. One of the great Girl Genius bits of dialogue results from this view: And there, for the moment, the matter rests. On the other hand, when Zeetha teases Agatha about her "boyfriend" and Agatha asks her to look after him, she doesn't bat an eye, nor does she show any particular hostility toward him (and Zeetha is not very good at hiding her emotions, all things considered). The discrepancy between these two views seems likely to result in a certain ... misunderstanding ... if Zeetha and Gil ever do run into each other.
 * Gil: "Kill me? What did I do?"
 * Klaus: "Absolutely nothing!"
 * Gil: "Father ... what did you do?"
 * Klaus (with an odd smile): "I kept you alive."

Other relationships
As far as can be told, Zeetha is single and unattached, as befits her status as Lost Princess. The Girl Genius Complete List of Absolutely Everybody! describes the large Yeti as her "boyfriend for a while," until she left the circus to take care of her Zumil. However, it's not really clear that their relationship consists of much more than Yeti schlepping her around after she sprains an ankle in the fight with the Monster Horse Beastie. (Of course, Zeetha is never one to let a sprained ankle get in the way of her responsibilities, say training Agatha.)

She seems generally well liked and gets along with most everybody except Klaus, although heaven help you if she finds out that you're a threat to her Zumil. The one time she turns her wrath on those around her for her own sake rather than Agatha's (see Yeti reference), you'd really have to say she's pretty well justified.