There seem to be a number of such tests, but no data collected from them.
I don't have a tulpa, and I tried the second test and was unable to keep track of both lines of dots; at best I could get one side perfectly and guess at the other side. If I create a tulpa at any point, I'll check if that result changes.
ETA: I tried the second test again, but counted the red ones as 1,2,3,... and the blue ones as A,B,C,... then I calculated what number the letter corresponded to. I got an almost perfect score; so a tulpa is not necessary to do well on this test. I'm not sure what sort of test could rule out this method; I have seen a auditory test which was two simultaneous dual-n-back tests.
Yup - since posting that comment I actually checked with the tulpa community and they referred me to those very links. No data formally collected, but anecdotally people with tulpas aren't reporting getting perfect scores.
I'm going with "use imagination, simulate personality" here, and am guessing any benefits relating to the tulpa are emotional and/or influencing what a person thinks about, rather than a separated neural network like what you'd get with a split brain or something.
The perceived inability to read the tulpa's mind and the seeming...
Thus spake Eliezer:
It seems that many here might have outlandish ideas for ways of improving our lives. For instance, a recent post advocated installing really bright lights as a way to boost alertness and productivity. We should not adopt such hacks into our dogma until we're pretty sure they work; however, one way of knowing whether a crazy idea works is to try implementing it, and you may have more ideas than you're planning to implement.
So: please post all such lifehack ideas! Even if you haven't tried them, even if they seem unlikely to work. Post them separately, unless some other way would be more appropriate. If you've tried some idea and it hasn't worked, it would be useful to post that too.