Our minds may be real things, but they're not part of reality or independent of it.
Real, but not part of reality? How do you parse that?
I think Brede is refering to a solipsistic view of reality, that your mind is the only reality, that perception is the only thing that exists, in comparison to the Objectivist notion that the mind is independent of reality, that just because you haven't perceived something, doesn't mean it doesn't exist, and that when you stop perceiving reality, it doesn't disappear, but your objectively existent brain that was previously generating subjective experiences just stops functioning, and everything else goes on.
It pretty much comes down to the fact that in order for there to be a mind, there needs to be a brain to generate it.
Consciousness cannot exist independent of an organ to generate it, and a physical object cannot exist independently of a physical universe with physical laws.
In order for LOA to be true, there either needs to be some fantastically complex and unknown physical mechanism for brains to be having massive effects external of the brain without use of the body (that for some reason isn't scientifically testable), OR you have to buy into metaphysical philosophy where our perception and thoughts can change reality on the basis of some inherent connection between reality and perception of reality.
Although the way Brede has worded it doesn't really make that clear (if this is what was meant), so apologies if I'm confusing the issue.