I’ve never been able to get very worked up about the terminology-based ones.  Is Cincinnati chili “real chili”?  Is Chicago deep-dish a “real pizza”?  I dunno, but they’re both good.
On the actual feuds, let’s see…
You can put clams in tomato soup, but I have no idea why you would, as neither the tomato nor the clams are improved by it.  The tomato tends to drown out the clams.  Much better to put the clams in a cream soup, where their flavor can shine through.
On every North vs. South feud save one, the South side is correct, and I say that as a loyal Yankee.  I will quibble slightly with JcWoman a bit about cornbread:  Sugar is allowed within ten miles of cornbread, if the bread is baked without it, and the sugar is in the form of honey, molasses, or maple syrup added topically to the bread.  It’s certainly not required, but it’s allowed.
The one point on which the South is just wrong is sweet tea, which takes the worst components of tea flavor, turns them up to 11, and throws the rest out.  Making the result so sickly sweet that you can stand a spoon up in it not only does not cover for this sin; it makes it worse.
Chili is best the way it was originally served, back when it was invented:  With beans.  Beef is good, too, but is not mandatory.  And yes, people have argued this point with me before, but even their own cites say that it was served with beans.
Apple pie with cheese is just weird.  I don’t care if others swear by it, but the flavors don’t complement each other at all.  A scoop of vanilla ice cream, however, while not necessary, is never unwelcome.
And the only stand I’ll take on the Barbecue Wars is that if you’re cooking, I’ll eat it.
Yeah, the only “two sides” I know of on that one are those who’ve never tried it, and those who love it.  We’re sort of the neutral arms dealer in the Hot Dog Wars:  Coney Island, Chicago, whatever, try some of this stuff on it.