Do you simmer with the lid on or off the pot? This is for tomato soup!

I think it depends on whether or not you want to reduce it, and of so, by how much. For tomato soup I would think you'd definitely want to reduce it while simmering, so I'd leave it either either fully or partially uncovered.

I would simmer with the ld off, or at most, with the ld on a little askew, as Sam1148 notes. If you cover (and seal) the pot, you might actually add more moisture than wanted, as moisture collects on the cover and drops back into soup. With a soup, this is not usually an issue, but can be an issue idctrying to reduce a sauce, etc. However, best advice would to be lid off.

My grandmother would simmer soups and stews, and say. "What you need to do is put the lid on like this; so it just sorta smiles at you" and put the lid on slightly askew. so it it did kinda 'smile'.

I think that holds heat in and some of moisture condenses from the askew lid back into the pot..and a portion escapes.

Simmer with lid off unless you have any vegetables within the soup that you want to soften. The simmering gently with the lid on can help make that happen faster

I usually simmer with the lid off, unless it's a stew or braise.