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She may be incontinant so I would reccommend having your vet check her out. Sometimes the sphinxter muscle stops working the way it should maybe because of longterm bowel movement problems, neurologic problems or possibly genetic's. It could also be if her stool is alittle softer and her anus was irritated at some point that she just not able to hold it in as well. At any case, I reccommend taking a stool sample and her into your vet to rule out anything serious.
Way to go Dr. V. !
It sounds like you are 'on the money'.
The only other thought that I could add is RE: anal gands in cats. Perhaps the staining is actually anal juice, especially if the cat is obese, for all of the same reasons that you have described. (I have often had to express anal glands in obese cats, especially long-haired varieties).