"The Duchess is a free(-)natured woman."
In the above sentence is there a nominal compound? Can we write "a free-natured woman" using a hyphen?
Are nominal compound and compound adjectives the same thing, and where is a hyphen appropriate?
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In the above sentence is there a nominal compound?
Not really. You could argue that "free-natured woman" was a compound of the adjectival compound "free-natured" and the noun "woman", but we don't normally consider such combinations of adjectives and nouns compounds unless the begin to have a meaning of their own other than what we can understand considering the adjective and noun individually. So that noun phrase isn't really a compound, and hence isn't really a nominal compound.
"Free-natured" is a compound, but it isn't a noun, so it isn't nominal.
Can we write "a free-natured woman" using a hyphen?
Yes. Indeed, I'd recommend you hyphenate adjectival phrases composed of two adjectives to make it clearer that the adjectival compound is a compound, and the noun-phrase it is part of is not.
Are nominal compound and compound adjectives the same thing,
No. Some people use nominal compound to mean a compound where the result is a noun. Some people only use nominal compound to mean a compound of two nouns, where the result is a noun.
Hence there is no overlap with compound adjectives.
and where is a hyphen appropriate?
This is a very big topic, because there are lots of cases where hyphens are always, usually or sometimes used along with a good few where people might use them but they are generally considered wrong. There's also considerable differences between styles.
Reading answers to questions with they hyphen tag is a good place to start.
When it comes to an adjectival compound formed from two adjectives, I would normally advise that you hyphenate, if in doubt, but there are exceptions.