I think the condition is that $D$ is diagonal and real, and $B$ symmetric. [Edit: forget that. It would be $\Lambda$ that the realness would apply to.] I am not certain though: when I wrote the earlier post, I wanted to ask a question about whether it was necessary and sufficient, but as I wrote, I realised I didn't know exactly what the question should be.
I saw you talking on the blog. A related issue is what happens when the process is not time reversible. At equilibrium, what is happening - what is cycling around and how? I've thought about that before, without achieving much. In the 4x4 case, you start thinking about 2 cycles and 3 cycles and figures of 8, just like Manoj discussed.