I call it "brook" because it's not a full on stream - it's just kinda there, tiny in the corner.
Commenting elsewhere, I pointed out one particular style element I'm trying to use in my book. When I first described it to myself (don't ask) I called it "writing from inside the character's head". Obviously missing that this is supposed to be the point of stream of consciousness, I just kept formulating my sentences in a way the character formulates sentences in his head. Still, the narration is in third person. Yes, it might feel an odd choice for such a personalised style, but I have good reasons for that.
So, when I'm "writing from inside the character's head", I'm not writing from inside one character's head all the time. I keep switching. Sometimes, when a situation calls for it, within the same sentence or paragraph. But more often, I point out how I wrote one particular fight scene. My main character notices a disturbing quality in the opponent and comments on it. And then, in the next paragraph, I switch to the enemy's point of view and write how he sees the protagonist. In the end, I had a couple of paragraphs when the points of view - and the emotions and sensations - switch between the characters all the time, in trying to show you how they both feel "in turns".
Some might find it odd, or bad, but writing it I liked to think of it as an experiment. This switching, "turn-based narrative" was something I wanted to try out and decided I like it. I think I will keep using it as I go on - unless some good, well argumented criticism proves me to be very wrong about using it.
Or maybe it's a good and commonly used strategy. Who knows, from my own experience it feels like something new, but I may be mistaken. Either way, for me it was an experiment that ended up favourably, for now.
It's not even a full stream of consciousness. I don't have, like, long ellipses and particles thrown all over the place. I just like to formulate my sentences like the thoughts inside the character's head, reflecting their feelings and sensations, while keeping the third person narrative.
Whether it ends up being successful or an unneeded experiment, time will tell.
Live long and prosper,