My answer is not from official rules of 5e,it's homebrew material (3.5 official though), however, I thought it could be useful since the author originally asked for ANY insight, it was the best answer I thought for a solution to the players playing ''one character''
There is a way to actually play 1 character with 2 players, if you want to try like I did back a few years.
(you'll both have a body but only one soul).
I played The 3.5 version of the Dvati race in the past with a DM, it was really fun, we each controlled one of them, but they are in synergy together, one soul but you can still make one a bit different than the other for some things like clothing, one can be more agressive but they always agree, with my DM we didn't want the others in game to know about how our race, so we did pretend to be 2 different persons. Plus since They can communicate via telepathy
at an unlimited range and across the planes,it was easy to act such a way.
The Dvati race (was official in 3.5) Source:
(Dragon compendium p.14, originally from Dragon #271)
Middle finger of Vecna however tried to adapt it to 5e, but is not an official source, but since you asked for ANY insight, I'm sharing it.