[Fyron]

Not that I can with 100% certainty state this, but I have some dault as to if such an anti-matter anti-clock device actually exists...
What is the theoretical background that indicates that it does exist.
Is it based upon observations or is it somekind of mathematical theory that states that it must exist in order to let ordinary clock's exist??

No. Anti-matter does not have to exist for matter to exist. Any specific matter item does not have to have an anti-matter equivalent to exist. Take your computer for an example. For an AM comp to have to exist for it to exist, then there would have to be AM humans that performed all of the exact same historical actions as us matter humans. They would have to have evolved in the same way, had all the same achievements at the same exact time, etc. This is of course so improbable to have happened that we can say with certainty that it could never have happened. So, this is why a matter item does not have to have an AM equivalent to exist. AM is not some mystical opposing force in the universe to matter. It is simply a theoretical class of particles that have opposing charges and spins to the equivalent matter particles.