Interestingly, I had this problem for 10 or more years, and last summer it went away. Now it returns occasionally. See the bottom for why. [no pun intended]
I believe it MIGHT have something to do with CONSTIPATION, or it's cousin "INCOMPLETE EMPTYING".
Constipation is by definition "being backed up" - and can occur due to MOTILITY being low, where our peristalsis is not adequate to the task.
Graphically - we poop, but the next movement's stools are already in the Sigmoid Curve, and we DO NOT manage to poop that out. Obviously, the poop is not supposed to sit there in the Sigmoid for long, and when it does we sweat a little bit out, just the brown stain but not stool.
PS - I am no expert except at observation - so I may be wrong in thinking the Sigmoid Curve is the last point before the exit. Whatever that final rest area is called, that is where our brown stain comes from.
WHY [from the top] - because I resolved my chronic constipation, at least for a few months. And yes, when I became constipated this autumn the brown stain returned.