Dear Sergios,

I am afraid this is not written up anywhere.
Briefly:

1). Let your alignment have two partitions - left and right.
2). SBP will infer two trees independently from each partition: LT and RT. There is also a 'joint' tree, JT, inferred from the entire alignment at the beginning.
3). A KH (or SH with just two trees) test can be used (for each partition separately) to test (by site permutation) whether the likelihood of a 'good' tree (topology and branch lengths) is significantly better than the likelihood of a 'bad' tree.
4). Run 1 option will run two KH tests:
   (a) For the left partition the good tree is LT and the bad is RT
   (b) For the right partition the good tree is RT and the bad is LT
5). Run 2 option will still run two tests, differing from Run 1 in what the 'bad' tree is:
   (a) For the left partition the good tree is LT and the bad is JT
   (b) For the right partition the good tree is RT and the bad is JT

Generally speaking, Run 2 option is going to be more conservative than Run 1, because JT has phylogenetic information from BOTH partitions, hence it should be (intuitively) less 'bad' than the tree inferred wholly from the other partition.

Cheers,
Sergei