Home made tomato soup? or canned?
With Campbell's soup there should be no problems....with home made it may or may not curdle....but to be safe add a little of the soup to the milk, and a little more and finally mix it all together....and after adding the milk, don't boil.
I usually don't add milk to home made tomato soup.
Linda c

Buzz it with a hand blender. As it heats all will smooth out. Have you ever served tomato soup with a splash of vodka and a dollop of sour cream?

I do as Linda does...add the milk to a bit of the soup in a measuring cup so as to temper it, then add it all into the soup pot. Low simmer, don't let it boil.

Maybe the information at the link below will help.

-Grainlady

Here is a link that might be useful: How to Stop Tomato Soup From Curdling

Yeahbut....those directions are for a can of concentrated soup...I mane mine from real tomatoes and celery and onions and don't want to use a stick blender, because it removes the chunks of vegetables which I love.
I don['t usually add milk ro cream but when I do it's by the tamper it method.
Linda C

I prefer my homemade tomato soup sans milk/cream.

But if I were going to make a creamy version, I would do it as Linda and Chase explained.

Linda

I make a creamy tomato bisque and simmer it with cream or milk and have never had a problem with it curdling. I also don't like to use the stick blender because I don't want a smooth texture.

Home Cookin Chapter: Recipes From Thibeault's Table

Creamy Tomato Bisque
====================
1/4 cup of butter
1 chopped shallot
2 stalks of celery chopped
3 to 4 tablespoons flour
1 to 2 cloves of garlic, cracked
Fresh chopped or canned tomatoes coarsely chopped
Chicken broth
Cream or milk
Bay leaf
Salt and pepper

Optional: Dill, chives or Basil
. Melt butter over medium heat. Add chopped shallots and celery and
saute for a few minutes to slightly soften. Add the flour and cook
stirring for a couple of minutes. Add chicken broth, cream (or milk),
the tomatoes, garlic cloves, bay leaf, season with salt and pepper and
simmer on low heat for 30 minutes or so.

Adjust seasoning to suit your own taste by adding fresh dill, chives or
fresh basil.

There is one exception. All of the above won't apply to canned soup. The directions on the cans are to add a can of water or milk and then heat it. There is no heating the soup before adding the liquid to temper as far as I can tell. Of course homemade is better, but canned soup is a quicker, easier option sometimes.

If you want to save some of the tomato pieces remove part of the soup, blend the rest with your milk and then add the other part back in.

If I'm adding milk to home made tomato soup, I make a light bechamel sauce and add that. This also adds a little "body" to the soup.

I heat my home canned tomato soup with a pinch of baking soda prior to adding the milk to the hot mix. No curdling.