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 alien 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.