Phrasal adjectives, the sets of words that combine to modify a noun, can be tricky. Usually, words in phrasal adjectives are hyphenated to signal their interrelationship, but there are exceptions, and confusion often arises when phrases resemble but do not constitute phrasal adjectives.
In English grammar, the assumption is that two adjacent nouns constitute a description of a single entity unless the first noun is hyphenated to a preceding adjective to form a phrasal adjective. For example, in “They earned their third straight regular season victory that day,” the reference appears to be to a season victory that is regular. Attaching the adjective regular to the noun season with a hyphen clarifies that the reference is to a victory that pertains to the regular season: “They earned their third straight regular-season victory that day.”
Why not hyphenate “third straight” as well? The phrase modifies “regular-season victory,” but “third straight” is not a phrasal adjective; the noun is the phrase “straight victory,” and third modifies the entire phrase (not just straight), which in turn modifies “regular-season victory.”
To see why this is so, replace the entire phrase — “third straight regular-season victory” — with any phrase beginning with an ordinal number followed by an adjective and then a noun (a phrasal adjective in place of “regular-season” is irrelevant): “first full sentence,” “second tall man,” and so on; no hyphenation is called for. The full sentence is the first one, and the tall man is the second one.
But isn’t this the same type of construction as seen in “The team achieved the third-highest score in the franchise’s history,” in which “third-highest” is correctly hyphenated because it modifies score? No. “Third highest score” refers to the third in a series of highest scores, but that’s not what this sentence is referring to; the reference is to a score that is third highest.
Also, in “The win snapped their opponents’ sixteen-game home winning streak,” “sixteen-game” correctly modifies “home winning streak.” But why isn’t “home winning” itself hyphenated to indicate that it’s a phrasal adjective modifying streak? Because it’s not a phrasal adjective. The modifier in this sentence is home, modifying the noun phrase “winning streak.”