Thanks so far. I also appreciate the above mentioned site, where it says:
As the center of mass moves farther and farther forward, you will need to dial in more and more nose-up trim to maintain the desired angle of attack. At some point the center of mass will move ahead of the center of lift of the main wing. The tail will then need to provide a negative amount of lift in order for the torques to be in balance, as shown in figure 6.5. There is nothing wrong with this; indeed most aircraft operate with negative tail lift most of the time.
As "most" doesn't mean "all", there must be some aircraft with positive tail lift?