I’m too lazy and tired to go into it at the moment, so I’m just going to paste this infographic explaining the relationship between the material base and ideological superstructure.

To the falsifiability point, while I can’t say a lot without knowing the specifics that Popper argued, historical materialism (and dialectical materialism, the way of understanding the world historical materialism comes from) don’t on the surface make much sense trying to attack from a falsifiability angle. While one could attempt to disprove, say, the extraction of surplus value through profit or the tendency of the rate of profit to fall being properties of capitalism (these are claims about the world that can conceivably be true or false), dialectical/historical materialism is the tool used to analyze the world, attempt to change the world based on the understanding from that analysis, incorporate the lessons learned from those attempts (be they failed or successful) into one’s understanding of the world, and repeat. It’s basically a way of gaining knowledge about the world, as well as an explanation of how people get knowledge.

Again, I’d have to check out Popper’s full argument for the specifics, but I don’t know how one can make assertions about the falsifiability of what is basically an epistemology without committing some kind of category error.