I enjoyed this.  Punctuation & spelling are good (there are
just a few typos scattered here and there).  The plot was
quite good, and your invention of the legend and its setting
is very imaginative.

Plot:   The story's strong point.  A few stock elements, but
        you sustain a pace which doesn't allow them to become
        tiresome.
Tone:   Conventional, but suits the plot well.
Characters:  I found myself in turns interested in what would
             happen to Aeka, Mihoshi, Nagi, Ryoko, and Kiyone.
             Washu is the only character you haven't brought
             to life.
Setting:  Interesting.  The planet Fantasia, the double-sun
          system, and the latter's planet were all very
          vivid.
Theme:  None detectable

This is based on
the TV Continuity - that is to say, post TMiL, but before Shin.

FYI, Shin Tenchi Muyo is a separate continuity.

Tenchi had acquired a taste for one of the bands
named Aerosmith.

You risk losing readers with digressions such as this.
It has nothing to do with your story; delete it.

set off a major earthquake that had partly leveled Tokyo.

Okayama isn't that close to Tokyo.  You probably should have
had the earthquake centered in Osaka or further south.

     "Who could blame you?"  Tenchi blinked...

This is confusing.  It appears to be Ryoko speaking here,
rather than Tenchi, so you shouldn't follow the quoted speech
with "Tenchi...".  Unquoted text in a paragraph containing speech
should relate to the actions of the speaker.  You do this
in a number of locations further down, as well.  Technically,
this is far and away the biggest barrier to reading your fic.

     "Emperor of Jurai Azusa is deathly ill.  It appears
to be only a matter of time before he will pass on.  He has
requested the presence of his daughters immediately."

Continuity nit:  if this is TV continuity, there is no
reigning Emperor Azusa.  In fact, the throne is either vacant
or occupied (remotely) by Aeka.  Aeka, Sasami, Youshou, and
Tenchi are the only surviving members of the royal family;
this is the cause of succession crisis which occupies the
second half of the first TV season.

That said, this isn't a serious flaw in your story.

     "Thanks for taking us, Kiyone.  We owe you."  She smiled.
    "I'll add it to your tab.  But are you sure it's wise to
bring Nagi with?  Especially in close quarters with Ryoko?"  Tenchi
sighed.
    "No.  But would you rather have her flying out there?  At
least in the Yagami, we can keep an eye on her; she did promise not
to bother Ryoko, and Ryoko is just trying to stay out of her way."
Kiyone frowned.
    "She agreed to that pretty fast."
    "I know.  But no sense asking why, as long as they aren't
trying to kill each other."  Kiyone nodded, and then sighed.

This is a good example of what I meant above.  You follow quoted
speech with the reaction of the _next_ speaker, rather than
the speaker who has just been quoted.  My reaction is irritation;
it's just very confusing to read, and takes longer than it needs
to, as I have to track back to check who's speaking.  This
is extremely non-standard, and I strongly recommend you change
to a more standard style.

    "First Mihoshi.  I did not know that it was you, Mother."

Whoa...  A _very_ unobvious interpretation of Mihoshi...
Intriguing.  At this point, I'm _very_ interested to see where
you go with this.

identity, and convinced the Marshal to say that he was my
grandfather." 

Another continuity nit:  TV continuity doesn't have a Marshal
as Mihoshi's grandfather.  (Of course, it doesn't say anything
contradictory, either.)

Even on Mars, you wouldn't have won if Ayeka hadn't distracted
Nagi.

Nit.  That episode was set on Venus.

...The name of her father."  Mihoshi
slowly nodded.
    "Garan, right?"  Nagi nodded.

So this is why you're using TV continuity...  (Certainly
wouldn't work with OAV)  But then again, you only need this to
give a reason for Nagi's feelings.  I suspect you'll hear some
complaints about the characterization you've given Nagi; your
version is interesting and compelling, but not, IMO, completely
consistent with the TV series character.

A long trail of gasses snaked
away from the larger, bluish star to swirl around the tiny white
dwarf in a whirlpool of shimmering gases and plasma.
<clip>
"The computer projects that in seven hours, both will go super-

nova."

A tiny white dwarf can't go supernova; its mass is below
Chandrasekhar's Limit.  It probably would have sufficed to use
radiation from sunspots or something like that to establish a
deadline.

     "Nagi is in the medlab.  Her right leg is broken; she passed
out on the way back."

The obvious thing to do at this point is give Nagi a leaf, as
Tenchi in fact thinks to do later.

     "My god... they aren't going supernova...  They're
colliding!" breathed Kiyone.  She turned back to the control board
and quickly back entering commands.
    "What's the difference?" asked Tenchi.
    "About a trillion gigatons!  This explosion will make a
double supernova look like a pop-gun next to Washu's dimensional
cannon!"

More flawed astrophysics.  Celestial phenomena can't change course
like that, and a stellar collision wouldn't be orders of magnitude
bigger than a supernova.  (If it were, the radiation would
sterilize the rest of the galaxy.)  You don't need this as a plot
device; you could have simply had Kiyone suddenly detect a flaw
in the calculations which led to the original prediction.

     "There's another ship!  And it's opening a hypergate!"
<clip>
    "What about the other ship?" asked Ryoko.
    "No signs of it...  Must not have made it in..."

We should have found out who was in this ship and
what it was doing there.  As it is, it's just a dangling deus
ex machina.


Summation and suggestions:

1)  Plotting is one of your strong points, but there is scope
for improvement.  Your major plot twists are unexpected, but
you resort to fairly stock situations for plot complications:
(supernova in 7 hours, Nagi's broken leg, etc).
Try to make _all_ plot twists unexpected.

2)  Your settings were good, but there is also room to make
them even better.  Think Tenchi OAVs:  imagine some really
unearthly landscapes, or planet-scale structures, then
describe them to us.

3)  Your twist on Mihoshi's origin and character is really
unexpected, and very interesting, but you didn't use the
concept after you'd introduced it.  You could do a lot with
this.  You also could have done more with Kiyone soul-
searching, after realizing how much she's underestimated
Mihoshi.

4)  Aeka's amnesia had potential, and certainly kept me
reading until it was resolved, but we didn't get to see
any permanent changes to her personality result from it.
There probably should have been a scene of genuine
friendship between Ryoko and Aeka towards the end, perhaps
similar to what's in TMIL2, aka Tenchi Forever.

5)  The Nagi/Ryoko relationship was explored very effectively,
and resolved very well (you could use this as a model for
how you might have treated Mihoshi-Kiyone and Ryoko-Aeka).
You tried hard to let us feel how difficult it was for Nagi 
to give up her hate for Ryoko, which is what you needed to
do.  There is room to get better; your goal as a writer
should be to make your readers feel Nagi's anguish to the
point where we're crying.  But you should feel proud of
the story as it stands.

6)  For your next story, you might want to try varying
tone a bit.  JMS once said that a successful TV script
includes at least one scene of everything:  one drop-dead
funny scene, one tear-jerker, one cliff-hanger suspense,
etc.

You are a good writer, and have the makings of a great one.
I'll be interested to see what you produce next.