How do we know that artificial shares of some stock aren't created (perhaps going untracked) on exchanges. After all being man made digital systems they are prone to errors.
An example:
Suppose a customer A and B,C meet at an exchange $E_1$ and B, C both want to buy a share of some stock S, and $E_1$ has a glitch that fills both of their orders using the same shares from A, now suppose (B,C) are high frequency traders, they could potentially execute an enormous volume of trades on other exchanges (which can't reasonably control for errors in $E_1$) and quickly this can break the consistency and accuracy of data across the entire market (also debugging the origin of the inconsistency could become extremely difficult as the problem spreads across multiple exchanges).
It seems to me that errors on any one exchange compromise the integrity the entire system, all it takes is one inconsistent player.