One of the biggest questions fans have about the Tenchi series, once they've done trying to figure out who Tenchi's going to choose, is what the crap is actually going on between Mihoshi and Kiyone?
Tenchi Muyo! isn't a big series for same sex relationships - and usually the focus is on the heterosexual nuances, particularly surrounding Tenchi. But Mihoshi and Kiyone - or Miho-Kiyo, as Washu coins them - are the one exception to the rule. And yet, there is evidence either way to suggest the relationship between these two.
My personal conclusion is that Mihoshi adores Kiyone, but she's not a lesbian and isn't romantically attracted to her in the sense that Ryoko is attracted to Tenchi, for example. Whilst she thinks the world of her, and relies on her in every single way you can think of, it's interesting to note that in Mihoshi's "World" in the Time and Space Adventures, her perfect world seems to be one in which she is married...to of all people, Tenchi. Whilst there is nothing in the TU series to suggest Mihoshi is in love with Tenchi, I think that the most important thing is that Mihoshi sees herself married. She wants a family and a child and to do that kind of thing. And though Kiyone is her partner, the one who she does everything with, she still sees that she could have a future with a man beyond their partnership at some time in the future.
Kiyone's perception of their relationship is different. In her world, she sees herself being pursued by Mihoshi, whose only desire is to hunt her down and be with her constantly, dragging her down deeper into a web of disaster and frustration. But it seems to me that Kiyone's own frustration is behind a lot of this self-pity, not Mihoshi's determination. Kiyone is often cruel to Mihoshi, as if trying to push her away and push blame onto her for things which are not always her fault. In a sense, it's almost like she's trying to assuage her own guilt in some way - that she's fonder of Mihoshi than she'd ever admit, and that she feels trapped by those feelings, because she doesn't really understand them.
Even in Mihoshi's world, Kiyone is not married, and when Mihoshi asks her about why, Kiyone is very defensive. In Kiyone's, it seems that Kiyo would give up everything - her career, her home, everything she has - if she could just escape the 'nightmare named Mihoshi'. However, when Mihoshi is stranded on the planet Earth, she is not worried about Kiyone or even thinking about her all that much. She is stranded, it's true, but she doesn't mention Kiyone at all until Kiyone comes to find her. In contrast, although she believes her dead and gone, Kiyone is still obsessing about Mihoshi aboard Yagami - even lighting a candle to her dead spirit in a weird kind of 'shrine' to her memory. Although she says that the loss of Mihoshi makes her happy, it's interesting that she goes to such lengths to track her down once there's a chance that her partner might be alive.
Of course, Kiyone's sense of duty is important in this too, because she isn't the kind of person who'd only half-obey an order or leave a job part done. But even so, considering how much she apparently hates Mihoshi, she still chooses to stick with her. And even though Kiyone is often frustrated by Mihoshi's carefree, absent-minded nature, sometimes you almost think that she's jealous of it - that she'd like to be as relaxed and happy about the world as Mihoshi is, with things not bogging her down. Perhaps she sees her partnership with Mihoshi as a way of becoming more carefree herself - that by being with Mihoshi, she might actually manage to see the brighter side of life. Unfortunately her rapport with her partner and her career prospects are often at odds with one another - so leading to much angst on Kiyone's part.
There are times later on when Kiyone adopts an almost maternal role towards Mihoshi, like she's someone she needs to protect. Since we see this from Kiyone also towards Mayuka, it suggests that Kiyone likes having someone to look after, even if she does occasionally throw a fit about Mihoshi's childish outlook. They often have disputes over buying snacks, going to karaoke, visiting the baths - but usually Mihoshi's ideas win through. And though Kiyone is frustrated by the number of jobs Mihoshi manages to lose them, they always take jobs together. Kiyone never tries to branch away from Mihoshi, even despite her complaining, and chooses not only to live with her in their own apartment (why, when everyone else is living at the Masaki house?) but to spend most of her free time hanging out with her, also.
From Mihoshi's side, Kiyone is the sister, the best friend and the confidant that she has always wanted, all in one. There is nothing she wouldn't do for Kiyone - even taking a bullet for her in Tenchi in Tokyo, despite the fact that Kiyone often - and loudly - blames her for any and all mistakes that happen around them. Mihoshi's innate sensitivity seems to understand that Kiyone needs her as well, in some way - and Mihoshi being Mihoshi, she's determined that she's always going to be there, no matter what. Mihoshi rarely snaps back at Kiyone about anything - only from time to time does she raise a protest - and her sunny disposition is probably the thing which, in the end, makes it impossible for Kiyone to stay mad at her. In fact, this might add to Kiyone's frustration, since it's pretty much impossible to have an argument with yourself!
We don't know whether Mihoshi in the Kiyone-related universes comes from a wealthy family like she does in the OVA, or whether they have influence in the Galaxy Police, so to suggest that any part of Miho-Kiyo's bond is based on hoped for patronage would be a slight on Kiyone's character (especially when the opposite is actually true).
Overall, it's my belief that Kiyone has more romantic feelings for Mihoshi than Mihoshi has for Kiyone, although that statement should not undermine the rapport Mihoshi feels for Kiyone or their actual closeness beneath their constant squabbles. Kiyone's confusion about her feelings and her subconscious fear that emotion/romance/relationship trauma is getting in the way of her career path is probably at the root of many of her problems, rather than Mihoshi's ability to do or say the wrong thing at the wrong time. All in all, if Kiyone took a step back and realised what she had in Mihoshi more often, I think she'd be more appreciative. But in order to reach that point, Kiyone has to accept things about herself and her own attitude/situation, and at the moment, she's just not ready to do that. So the blame keeps falling on Mihoshi's oblivious head - and on they go as ever.
While I wouldn't call them an out and out lesbian couple, on account of Mihoshi's 'inner thoughts', I think they're probably as close as Tenchi comes to having one. Noboyuki actually comments that a "close friendship between two women is a beautiful thing" and considering the kind of guy he is, that's a pretty loaded statement full of implication. There's no doubt in my mind that Kiyone is far more involved emotionally with Mihoshi than she will ever be with Tenchi. Which is something of a relief, actually...