It is certainly a gray area: neither phrasing is entirely disallowed. "Drink" is always associated with liquids, as you know, and "eat" is typically associated with solids. In addition, "drink" is usually what you would do to relieve thirst, while "eat" is what you would do to relieve hunger. Another way to look at it is that "drinking" is usually associated with beverages, while "eating" is usually associated with foods. Also, anything done with utensils is more likely to be thought of as "eating", whereas a liquid consumed directly from its container would be more likely to be thought of as "drinking".
None of those points decides the issue categorically, I think. However, we can use them to narrow the issue: anything that is mostly a solid would be eaten, while anything that is considered a beverage would be drunk.
Personally, I (native AmE speaker) prefer "eat soup" over "drink soup" regardless of the composition of the soup: I would consider it food (and not a beverage), and I would think of it as predominately something to alleviate hunger, not thirst (although obviously it could do both). If the soup is not entirely liquid this preference is stronger. Likewise, if the soup is eaten with spoon, as opposed to directly from (say) a mug, my preference for "eat" is stronger.
For context, I would have much less of a preference for which verb to use with broth: my verb choice would probably depend upon how the broth was consumed. This is probably because I think broth is closer to the line between food and beverage. On the other hand, I would probably always say "drink a smoothie", even though it might have more food value, simply because I think of it more as a beverage.
The OED has these relevant definitions of "drink": "To take (liquid) into the stomach", "To swallow down the contents of (a cup or vessel)", "To swallow down (something solid) in a liquid", To swallow down or imbibe water or other liquid, for nourishment or quenching of thirst". These seem to leave the question of soup open; the "swallow something solid" definition seems to be referring more to incidental solids (such as a bug in one's drink). I do notice that none of the quotes under drink refer to soup.
Under "eat": "To take into the mouth piecemeal, and masticate and swallow as food; to consume as food. Usually of solids only." and "Of liquid or semifluid food. Now chiefly with reference to soup, or other similar food for which a spoon is used." This seems to support the idea of the utensil distinction.
Google has 376,000 hits for "eat soup", and 97,500 hits for "drink soup". Of those, the entire first page for "drink soup" seems to be discussion of which term is better, while there are three such hits on "eat soup".
Summary: I personally lean toward "eat soup" in all circumstances, but would not react negatively to others' use of "drink soup". I would strongly suggesting using "eat" if A) the food is largely solid, regardless of what it is called, B) it is consumed with a utensil (either generally, or in the particular case being discussed) and C) it is considered "food" (as opposed to "beverage").