Conduct a search for identical twins raised in the exact same environment. Then after finding some, scientifically investigate what the differences are, say in IQ, musical ability, any kind of ability, and any other way in which the twins could test differently or the same.
The results will indicate if individuals with the same DNA and the same upbringing can differ in anyway. If differences are found, then you have to hypothesize on what gave rise to the differences as both nature (exact) and nurture (approximate) were identical.
One hypothesis is that the differences derive from different souls in different reincarnation lineages occupying either twin body. On the basis of this hypothesis any differences may be attributed to a different set of reincarnation experiences. The data will inform as to what characteristics come from nature plus nurture and what characteristics come from reincarnation.
For example, the two twins should be distinguished by there respective karma, so that in the present life they will need unique experiences to balance the karma. So the twins should be distinguished by what each is driven by their inner motives to experience in life. This is a testable hypothesis. Inner desires should be dependent on karma derived from past lives.
For another example, I am supposedly the reincarnation of Clem, a famous boy drummer in the Civil War that killed a Confederate Colonel and went on to become a Major-General. However, he always wanted to be a drummer but could not because it would not be becoming for a rising star soldier to drum. He died in early 1937 and I was born in late 1937. I got appointed to West Point by the Grandfather of our president, but poor eyesight kept me out. So instead I worked for the Army on Star Wars. But my real passion was colonial drumming in the Boston area. I just came back from a colonial faire and fife & drum muster where I did not sleep but played the drums day and night all weekend. So this time around I am satisfying my karmic need to drum.
So differences in the life history of identical twins should have some dependence on past karma and should be evidence for reincarnation.
While on this subject- for any who have read this far- reincarnation is not an escape from death. You will have continious consciousness after the physical death, but inorder to be born again, you need to lose all memories of your past lives. That is equivalent to ceasing to be alive. It's called the second death in scripture. So the period between lives is just a preparation for being born again. There is some thought that during this period you can remember all your past lives, but I do not know how to test for that.