I use 2 acoustic guitars, one with a capo.  Play the same chords, but with inversions.  In particular, i've started putting my capo on the 2nd fret, but leaving the low E open (offset the capo so it only holds the 5 other strings).   Because of the keys I tend to write in, that low drone can really add something special but doesn't affect any of the other standard fingerings on the notes above the 2nd fret.
Not exactly an alternate tuning, but relevant.