I'm trying to improve on Inspector De Rosa's idiopathic acute systemic inflammation. Originally, he was coughing all the time, which was too annoying. Now he almost never has a flareup, which is too little. It's actually important later in the story, so I'm thinking of making it psychologically/supernaturally induced, with flareups/stigmata whenever he's confronted with grave sin, including the crucifixions of the cardinals. I could have another flareup when Connor and Alessandro are struck by lightning. If I go with stigmata, he'd have to wrap his wrists a lot to hide it/keep from bleeding over everything. The neck wound could be in the shape of the cross to match the one he always wears. The bloody cough would remain unchanged, and I'd add a raw throat. He's going to need a lot of handkerchiefs and uniform shirts. :-)
In real life, a condition called hematohidrosis is thought to be a possible cause of stigmata. Basically, people sweat blood in response to intense emotions. The Bible mentions that Jesus's sweat was like blood while praying in Gethsemane before his arrest. Another possible cause is malignant malaria, which causes bleeding from the hands and feet. St. Francis of Assisi is thought to have suffered from that.
EDIT: Another alternative is to go with full stigmata (hands/wrists, feet, side, and forehead) and get rid of the bleeding neck. I'd keep the bloody cough, though, as I want him to suffer, not just bleed externally.