Welcome to ParaView, Ben!
The problem is that you are using the ‘Volume’ representation on two overlapping volumes, and that is not supported in ParaView. You won’t get a warning, but it will just look weird, as you have noticed.
My recommendation is to extract a contour of the vessels and volume render the cells. Use the Contour filter to extract the vessels. Intermixed surface and volume rendering is supported, and should give you the effect you are looking for, e.g.,
