How to get Solid union effect and still reference each brep separately

I will use two intersecting pipes (cylinders) as an example. We have a vertical and a horizontal cylinder:

By using solid union, we can construct their intersection properly. But the result will be one large Brep. Is it possible to have the same effect while also maintaining separate Breps for the horizontal and vertical cylinder?

Have you tried Solid Difference?

Horizontal cylinder - vertical cylinder…