Look at any Kruskal–Szekeres coordinate plot of the Schwarzschild solution. It shows the same mass everywhere. Yet the two sides cannot talk to each other, in that no information, particles, etc can cross the wormhole throat. So how do the sides 'know' to be the same mass?
So is there a way to draw a Kruskal–Szekeres plot with the masses unequal on each side? In other words, would the geometry of space play nice and smooth at the interface between regions II and III, where different mass solutions are right next to each other?
Another way of putting this is that if you overlap two K-S diagrams with different mass, M1 in region I && II, and M2 in region III and IV, and then do an embedding diagram, will you see something different than the single mass version.
Another way of putting it. The Schwarzschild solution is static, and unique. So can you sew two of them with dissimilar masses together coherently? If not, then it would seem that another - non static - solution is in order, which would be surprising, since there is only one parameter (M) to be non static.
Look at say http://www.csun.edu/~vcmth00m/embedding.pdf or similar for diagrams.