"What do you eat with fish?" would be "Що ви їсте з рибою?"
"What do you eat fish with?" -> "З чим ви їсте рибу?"
Slightly different, even though the meaning is basically the same x)
З чим ви їсте рибу? - З лимоном. What do you eat fish with? - With lemon (juice))
Що ви їсте з рибою? - Картоплю. What do you eat with fish? - Potatoes.
I am a native Ukrainian, but it seems that "the" should not be there. You are asked about eating fish in general.
With what are you eating the fish? -- asking about some particular fish that you are eating now
With what do you eat fish? -- in general, asking about your preference
I wonder if Ukrainians may misunderstand this question not as "With what garnish do you eat fish?" (e.g. with chips) but as "How / with what means do you eat fish?" (e.g. with a fork and knife)? I've always supposed this funny conundrum does not occur in inflectional languages, but now I've seen an example of one, I start having doubts. I guess the later meaning is implied by "Чим ви їсте рибу?", but I'd be thankful if a native clarifies the issue...