A tree with no convergence would have a consistency index of 1. How you calculate CI will determine whether you get a high consistency index or not with unresolved branches - how are you handling the lack of resolution? Basically, it’s an apples and oranges problem - a CI value for a tree that is not completely bifurcating is not measuring the same thing as a tree that is completely bifurcating, just as comparing tree statistics generated by different data sets is not generally very meaningful. What version of PAUP are you using? In the current PAUP* 4.0a168 (beta testing), if you scroll down the analysis menu, bootstrap/jackknife gives you ways to test the strength of groupings in the tree through different resampling strategies. Note that likelihood analyses are not too fast to begin with and the bootstrap or jackknife involves repeated analyses, so you need to allow a fair amount of time for it to give reasonable numbers. If you want to try parsimony analyses, TNT analyzes the data faster but is not as user-friendly - you’d want to build your file in PAUP and then analyze it with TNT.
Note that running PAUP*4, using your data set of
Dimensions ntax=4 nchar=2;
Format datatype=DNA gap=- missing=X;
a branch and bound search gives three trees with a CI of 1 (describe trees) but also it lists a CI excluding uninformative characters, and that CI is zero. The CI excluding uninformative characters is what is meaningful for assessing the type of question you are asking. Any data set where none of the examples actually share any features will automatically give a CI of 1, but none of those features tell any sort of phylogenetic analysis anything meaningful - there needs to be some commonalities to work with.
An appropriate comparison would be analyzing a data set made of several long random strings of characters - for convenience, say A, G, T, and C, and see how the results for that data set compares with a set of actual DNA sequences that are appropriate for a particular group of organisms. (By appropriate, I mean that it needs to fall somewhere between having essentially no change, which again would give high CI but be highly uninformative and having so much change that it is essentially randomized. As DNA has only A, G, T, and C to choose from, a sequence that mutates enough will have random matches with other sequences.)
The actual patterns observed with analyzing data for real organisms closely matches the expectations of an evolutionary pattern, which is to have pretty good nested clades but also some convergence, random variation, and other “noise” [from the point of view of someone trying to figure out the evolutionary relationships, those are noise, but not necessarily to someone asking other types of questions.]