This is a strange one. Since the alternator is new I presume all the wiring to it is good and connected right. This looks reminiscent of the sort of weirdness that happens when you get a bad earth. All I can suggest is check all the negative connections to engine, body and chassis (and batteries) in the engine bay. With the engine ticking over, when you put a multimeter between engine block and a positive battery terminal are you seeing 14v-ish?
Btw did you get to the bottom of your problem where it was slow to start when cold?
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