A BSP tree can only work in normal space because of how the tree is built, but if you have multiple BSP trees that overlap then it's more like 5D space.
I know how to organize overlapping 5D space into normal spaces, it's what I call sector groups.
Sector groups separate overlapping geometry into normal spaces.
How it works is when building BSP trees the different groups represent separate BSP trees.
There are sectors that belong in two or more groups and are included in other separate groups when building BSP trees.
The portal system brings all the separate BSP trees together.
A map file that contains the geometry and separate BSP trees work with a portal system.
The portal system would have to have the sectors and separate BSP trees in one file.
The BSP trees could make sectors and portals automatically, but need some helper portal to tell that a portal from tree 1 connects to a sector in tree 2 for the portal system.
I don't completely know if this will work with BSP trees, but I have been experimenting with sector groups and portals in Unity.
Sector groups work great in Unity to stop the player from falling out of the map.
When the player crosses a portal, the mesh for group 2 is turned on and group 1 is turned off or the player collision layer changes from group 1 to group 2.
The portal system would need some kind of portal clipping to work properly.
What I am saying is overlapping 5D space can be put into separate normal spaces and connected with a portal system.
The separate BSP trees are used for collision and searching for sectors.
I know transformation teleporting portals are used with BSP trees, but I want to try this with visibility portals.