For me (and I suspect maybe other Brits that have voted), "other" is brown sauce. The idea of any sausage on bread of any nature - a cob, a sandwich, a hot dog - WITHOUT brown sauce, is ***** to me.
Onions, one line of brown sauce, one line of lovely yellow mustard. Sharp and sweet on the same sausage.
Incidentally while we don't really do chilli dogs over here, I understand the appeal. I've had chilli on a burger before. But I imagine the idea of throwing a sauce on top of ready made chilli just to result in a confusing mess. If I want to throw chilli on top of my meat, it's because I want it to taste like chilli. Not mustard, relish or mayo.