My bet is on your battery being so dead or nearly dead that it's not putting out enough power to provide the alternator its required +12v exciter voltage. The alternator uses 12v from the battery to power its circuitry inside, at least on the type of alternator used in our Wranglers.
Like if you entirely removed the battery while the engine was running, which you should not do, the engine will eventually die. A good charged up battery is required for the engine to run properly.
When you have a choice, buy American.