Do I need to put “the” before “most” in this sentence?

Is putting “the” before “most” in this sentence compulsory, optional, or a mistake?

Fascination with language and attention to particular regions and communities in America are the most common themes for which Coen brothers’ works are appraised.

If it is purely optional, how does its presence or lack change the tone of the sentence, in case it does?

Short answer: "The most" is correct here.

Long answer: Most can be used as both a superlative and an intensifier. You are interested in the superlative use: of all the themes under consideration, these are the ones that are present in the greatest number. In formal usage, you would almost always want to use the definite article the with a superlative (except for possessive constructions like "John's youngest child").

In practice, the is frequently omitted but implied (e.g., "I like this one [the] best"), which complicates the rule somewhat. Nonetheless, if you follow the rule you'll almost always be right.

The other usage of most that concerns us here is as a somewhat archaic intensifier, as in Shakespeare's "murder most foul." In this case, most is being used as a synonym for very, so it doesn't take the definite article. Unless you habitually wear spats and a monocle to dinner, you'll probably want to go easy on this use of most.

With the "the", you are saying that among all the common themes, these are the most common -- the top-ranked ones on the scale of common-ness. Without the "the", you are saying they're pretty common but there might be others just as common; technically "most" doesn't add anything, but functionally it adds some emphasis.