VTK_POLY_VERTEX rendering issue

I have a rendering issue with VTK_POLY_VERTEX cells. In my situation the vertex cells have enlarged point sizes, set via Actor->GetProperty()->SetPointSize(14).

The vertex cells are placed inside a box mesh, i.e. they are completely enclosed in the mesh. However, under certain camera angles the vertex cells shine through the box. How can I control this, i.e. where in the renderer/actor/mapping settings can I switch this behaviour off?

Here’s an example with 3 polyvertex cells shining through - one fully, two partially.