Well... nothing much would happen, I imagine. We could only join with Ardbert, because he was already dead, but was kept from rejoining the life-stream of the First by Hydaelyn's direct intervention, allowing him to join with us later. With Hydaelyn gone, the chances of that happening again are pretty much zero.

What is a more interesting question to me is, what happens to those who traveled from one shard to another and died over there? Like the native Alexandrians, who came to the Source. Does their soul somehow go back to their own shard, since it still exists in some form or another, or does it stay in the Source?
And if it's the latter, what does that mean?
If one of the two dies, do their souls merge, or does the dead's soul wait for the living one to also die to rejoin, or is it just going to be reborn as a seperate being, until both parts of the soul end up being dead at the same time? Or does the lack of a proper rejoining mean, they'll never re-fuse and will stay seperate?