The white hole and black hole regions in a Kruskal diagram are said to be actually two different locations. Given the problems with white holes it might be a silly question but, hypothetically, what would a stationary observer in the $r>2M$ region supposedly observe? Surely not two spheres, one white and one black. I imagine they would only see a white sphere.
(I know 'matter can't come from nowhere', particularly the $r=0$ singularity in the white hole region, but just suppose the latter was 'covered up' somehow and there were torches or stars in there).
So I imagine 'spatially' an observer would locate them both by pointing to a sphere, but the white hole would be what was there 'in the past', the light they see is coming from there; whereas the BH is what they would encounter, in the future, if they decided to jump in. Notice, up until the observer crosses the future horizon they still think they are headed towards the light they see coming out of the white hole. Only once they enter will this cease (unless of course there are also stars and light coming from the other asymptotically flat space).