the phrase has both examples of the use of the word de.

what is she eating.. answer could be a piece(un); the piece(de +le=du); some(implied)pieces(de +les=des)... hence des petits morceaux.

Can you tell/describe the kind of pieces? .yes 'tomato pieces' (noun (tomato)acting as adjective).. hence the invariable 'de' and the singular tomate without the article because it is acting like an adjective here.

If one had said , she likes to eat lettuce and tomatoes it would have been...manger  'de la lettuce' and 'des tomates', if I am not mistaken.

But I came here to ask why "Elle s'appelle Caroline" is the only accepted answer. Notwithstanding that 's'appeler'  is the term for people, the turtle can never 'call herself Caroline' really. Wouldn't it be better to say  "Caroline est le nom de ma tortue".

I think the rule about 'des' becoming 'de' before an adjective + noun is a rule for formal or written French, and not how people speak.