Can you really pull a battery cable and the engine still runs?

I'm not sure you can, at least with an alternator with a field coil. The alternator needs a DC voltage source to run a current through a coil that create a magnetic field inside the alternator so that the rotation of the alternator around that magnetic field creates current. That DC magnetic field comes from the field coil, and the voltage source for the field coil comes from the battery itself.

One thing you don't want to do is start it with a battery and disconnect the battery. If you do that, the alternator has been dumping current into the battery, and when you disconnect the battery, that current has nowhere to go, and ends up with a voltage spike that can destroy your vehicles electronics if they are not protected. This is called a load dump, you can google it, I'm not making it up.

I could be wrong...I know electronics, cars maybe not so much. I could be missing something about how a particular car works, like an alternator with a permanent magnet or something.