You should be able to unearth the soup with a partial match. What makes the alcohol/alcohol-free problem is that alcohol free is one token that can't be subdivided.
Oh! Chicken and rice soup, I could go for that. Cart.
Moving the argument into the realm of things that matter, chicken and rice soup is not a product that can be called "rice", and therefore isn't an appropriate response to a search for "rice". The title is irrelevant; if you're stocking rice with a fancy name and its title doesn't include the word "rice", that product still should show up in a search for "rice".
Yes: chicken and rice, rice cakes, rice paper, cauliflower rice, rice cooker, rice krispies, rice university jersey, mochi, risotto.
No: licorice, ricetta tiramisu.
As I said, that is inconsistent with your simultaneous position that alcohol-free wine is not a match for "alcohol". Either they both are matches to their respective queries or they both aren't; there isn't a difference between the two examples. This is true from the perspective of dumb text searches, but it's equally true from the perspective of correct linguistic parsing.
That rice is part of the listed title for chicken and rice soup is a coincidence; the argument for returning it also tells you that you should return the same soup when asked for "tomatoes", if tomatoes are an ingredient in the soup.
Your "yes" and "no" examples don't seem to work. What rule excludes "licorice" that would include anything containing the raw textual string "rice" without regard to whether any rice is involved? Rice paper, rice cooker, cauliflower rice are all purely textual matches. Mochi is a purely semantic match!
It's not even a problem... it's an unfounded assumption that alcohol-free wine wouldn't ever be relevant to the user at all. If the only place open on a holiday was Walmart that I didn't realize didn't sell booze, and I needed some kind of booze to make a pan sauce or a braise, I wouldn't necessarily think to search for non-alcoholic booze-- but it would be a very useful substitution for me, and not by accident... they're two versions of the same product. I'd much rather see that they had non-alcoholic beer and wine in my search than completely empty search results.
This is why competent companies hire designers to work on interactions like this rather than leaving it up to the implementing developer's gut instinct about user needs. If only the tagging could keep up.