So, for a regular major chord, you would, for instance, have your open E-Major chord and have the second strings superimpose a B Major chord over it. So, you would have the notes:
E - G# - B - D# - F#
Which in essence would be a EMaj7 chord with a Major 9th added to it, aka Emaj9. All these notes fit in the E Major scale so it should be a pretty swell complex chord with no apparent clashes. Harmonically speaking, pretty much anywhere an Emaj7 fits, an Emaj9 works as well.
Now, if we take an E minor shape, the second strings would now superimpose a B minor chord over the Em, so you would have the notes:
E - G - B - D - F#
Now it seems like we have in essence a minor seventh chord with the Major 9th added again: An Em9 chord. No clashes here again as all these notes fit in the e minor scale. Em9 chords sound quite nice, and function essentially like an Em7 chord. Notice, though, that as one adds more extensions to the chords, the chords are diatonic to fewer and fewer keys (as an example, Em7 is diatonic to C major, but Em9 is not).
If we play our A minor seventh open chord shape, we would now have a A minor seventh with an e minor seventh super imposed on it by the second strings, so you would have the notes:
A - C - E - G - B - D
Here you would have a minor seventh chord with a Major 9th added and a perfect 11th or compound perfect 4th. This is an Am11 chord. You may be seeing now that these strings are going to add a second chord built on a root a perfect 5th higher, and of the same chord quality, to whatever you play. Am11's a cool chord to play, but that's a lot of notes. You might not want that many notes, because you might be sort of smothering the harmonic space. Keep that in mind.
If we use our open A Major seventh shape it would be that chord with a EMaj7 super imposed on it by the second set of strings, so we would have the notes.
A - C# - E - G# - B - D#
This would be the AMaj7 chord with a Major 9th added and a compound augmented 4th or an augmented 11th. We call that Amaj9♯11. Try using this chord in E major, or if you're feeling a little adventurous, A major, since this is a Lydian chord (it implies a Lydian tonality) due to the presence of the D♯.
Here we would have, for instance, an A dominant 7th with an e dominant 7th super imposed on it by the second set of strings, so you would have the notes.
A - C# - E - G - G# - B - D
This is probably the only one of the chords that you would find will not work as it seems that there is a clash between the minor 7th of the A chord and the Major 3rd of the E chord. It's a nasty, crunchy sound, and you'll likely find it very difficult to actually play any dominant seventh chords because of this. That's probably a huge problem right there, since lots of music uses dominant seventh chords.
If anyone wants to take a stab at what the minor 7th flat 5 would be or the diminished and half-diminished 7th chords would be, I have made this a wiki entry so we can have some fun with the theory implications.
If you wanted to play extended chords,
E-G-B-D-F♯ becomes E-G-B-D-F♯-A-C♯
That's Em13. We already know dominant chords are a bad idea, so Emaj9:
E-G♯-B-D♯-F♯ becomes E-G♯-B-D♯-F♯-A♯-C♯.
That's Emaj13♯11, hardly surprising. The reason most of these larger chords become comparatively small chord symbols is because the maj7 and m7 chords are two distinct perfect 5ths. However, the 7 chord is not, so it produces a clash with its P5-higher duplicate.
Suspended chords could be cool, as well (I'm going to stop running through every note). Esus becomes Esus4sus2 (or whatever you want to call it). E7sus also becomes Esus4sus2. E9sus gets more interesting, becoming [E-A-B-D-F♯-C♯]. I'm not going to attempt naming that as an E chord, but it does make a Bm11 chord, which if you think about it makes sense, since m11 chords can be voiced as quintal or quartal stacks. E13sus becomes [E-A-B-D-F♯-C♯-G♯]. That technically forms an Emaj13 chord, but probably not one you'd want to use. More useful are the stack-of-fifths chords you can make out of this tuning.
And trivially, power chords become (1-5-9) voicings. Those could be pretty cool. Playing an octave would make a power chord (actually, playing any note does the same).
Melodically, this tuning could be really cool. It would essentially harmonise everything you play a perfect fifth higher, and I'm pretty sure there are pedals out there that do the exact same thing. Almost every note in most scales can be harmonised a perfect fifth higher, and even the one that doesn't work tends to just get glossed over anyway (scale degree 7 in major). It's already a common sound in metal to have a power chord on the vii degree, and I'm sure it wouldn't be unwelcome in other music.
The net effect of such a guitar is that most of your tertian chords are going to come out of this guitar as much bigger, fuller sounds. However, the downside there is that you then have to be cognisant of all the crazy extensions that get thrown into whatever you're playing. And not having any dominant sevenths available is probably a bad thing. You might see some use for making massive-sounding voicings of quintal/quartal chords, but functional harmony is going to be difficult to emulate. Perhaps you'd want to change the tuning such that only some of the strings are doubled a perfect 5th higher?
Another thing to consider is that the main reason this tuning is hard to utilise is that it messes with the harmony that the guitar involves itself with. In order to explore the idea of using these fifth-paired strings, it actually might suit one to do this tuning setup on an 8-string bass rather than a 12-string guitar.
The reasoning there is simple: The bass in music typically is not responsible for playing chords and implying large, dense harmonies all at the same time like a guitar does. The bass role in music is a lot more melodic in nature, and usually plays one or two notes at a time. It would be a pretty cool effect to have every note played on a bass be largely the same as a power chord, allowing for ridiculously fast power-chord-like riffs in the lowest registers. I could easily imagine this setup used in metal and rock, where guitar and bass riffs are very important parts of the song.
This is actually an effect that can be done on a regular guitar or bass with a pedal, similar to the concept of an octave pedal but with a fifth. The perfect fifth above any note is nearly always a part of the same key, and even when it isn't, power chords on notes that technically shouldn't be harmonised a fifth above (diatonically speaking, anyway) is already kind of an accepted sound in a lot of rock contexts. The whole concept of a power chord is basically one note that sounds bigger and stronger, so I think having these automatic power-chord-like textures on everything could be a very interesting idea.
The other nice thing about fifths is that they don't tend to sound as muddy in the low register as most other intervals. Obviously, thirds tend to get muddy down on bass guitar, but fifths don't tend to violate the lower interval limit nearly as often due to their simple ratio of 3:2. Octaves are the only option that's less muddy, and of course regular 8-string basses use octaves for their paired strings with zero mud at all.
And finally, there's an effect called subharmonics that basically says that by combining a note and a note a fifth above it, you can actually hear a note an octave below the lower of the two notes you're playing. You could even observe this by adding two sine waves a fifth apart and looking at the pattern you end up with, which will have a wavelength twice as long, which represents the subharmonic note that the brain perceives. This is very similar (actually, it might be the same thing) to what organists use to reach notes far below the ends of their keyboards. On an electric bass, this could be experimented with via amp settings and the like, and you could easily figure out how to really get a great tone for these notes.
Imagine: Lightning-fast power chords, devastatingly low riffs, and the potential for zeroth-octave notes without nasty drop tunings. This fifth-based tuning for the electric bass could be revolutionary for music if even some of the above takes off.