Joe,
Noting special on this. I did "confirm" a complete fusion at the bottom of the vase from which the two branches emanate. I always look for that and make the tenon in that solid section.
If that area is cracked, weak, or not solid, I have spanned the joint with a waste board for more aggressive parts of the turning. For the initial drilling/coring I have, at times added a waste board to the top and then turned/bored the ID through the wast board. I have also made an epoxy pour into the bottom after the center is drilled/cored for stabilizing the vase if needed.
Hope that makes sense. Maybe I need to offer a vase class?