"Bloody show" is a colorful term for the mixture of blood and mucus that indicates the cervix is preparing for labor.
It is one of the many strange, sticky bodily secretions you are likely to encounter for the first time when you become a parent: colostrum (yellowish early breast milk), meconium (baby's tar-like first bowel movement), vernix (the creamy white stuff often covering newborns when they're born — also known, nauseatingly, as "birthing custard").
I thought of the term again recently as I watched an episode of "This Is Going to Hurt," a darkly comic TV series following a team of stressed-out doctors and midwives in the obstetrics and gynecology ward of an under-resourced London hospital.