Modern, sophisticated dualists tend to think that consciousness just sits "on top" of brain activity, experiences are generated based on what's going on the brain. Many are "epiphenomenalists," who say that the content is determined only by brain activity, so the information flow is one way: brain -> conscious experience.

This has loads of problems (if consciousness has some influence over behavior, then what's the physical mechanism? And if there's none, then why are we able to even talk about it?). But dualists argue that materialism is even worse because it leaves you with outright paradoxes.

So, on this account, a weakly conscious system is just a system that's physically organized in a way that generates only weak experiences.