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I'm confused about how to combine an open-form compound word with a word that would normally be hyphenated. There's excellent guidance for making the open vs. closed vs. hyphenated decision, but I don't see how to apply this when hyphenating the open-form word looks wrong.

For example, make a compound word out of North, America, and based. North America is open formed and something-based is hyphenated. Is Coca-Cola a...

North America-based company: this seems very wrong as it de-emphasizes North America as a proper-noun place and makes it sound like the company is based in the North part of America (which is neither accurate nor the intent of the phrase).

North America based company: feels jolting to read and omits what seems like a necessary hyphen before "based"

North-America-based company: looks best(?), but has hyphenated the open-formed compound "North America", which unlike "well-thought-out plan" still seems wrong, despite the guidance at the linked answer above regarding phrasal adjectives*.

_{* the aforelinked answer says every word is hyphenated in phrasal adjectives , but for some open-form words this looks wrong}

Note: I think my question could be improved with an example that looks even more egregious, but I can't think of one.

The Chicago Manual prefers a spare hyphenation style; their guideline is "hyphenate only if doing so will aid readability". So Chicago would recommend North America based.

When I look up based in Wordnik, all of their examples where based is preceded by a proper name use the hyphen, e.g., U.S.-based, N.Y.-based, and so North America-based by extension.

However, I would share your reservations about joining America to based, and would use North America based.

The Chicago Manual notes:

Far and away the most common spelling questions for writers and
editors concern compound terms—whether to spell as two words,
hyphenate, or close up as a single word.

To aid your decision, they offer this handy table.

One thing some style manuals suggest in this case is to use an en-dash rather than a hyphen. So

North America–based company

rather than

North America-based company.

The longer dash signals that it shouldn't be parsed as "America-based".

Based on a cursory scan of Google Books for North America based, where their search engine ignores any punctuation marks between the words, I would guess that about 2/3rds of all relevant instances were North America-based. But I see nothing wrong with omitting the hyphen.

I didn't see a single instance of OP's doubly-hyphenated version, which looks decidely odd to me.

Rewrite the sentence to avoid the problem:
Coca-Cola, based in North America, makes sugared water.
Coca-Cola, headquartered in North America, makes sugary water.

Or just drop "based"
North America's Coca-Cola makes sugar-water.

My convention is that I hyphenate if the term modifies the following noun, so "North American-based company" is correct.

My related convention is that if the modified noun precedes the -based language, I remove the hyphen: "the company is North American based." This is consistent with the Chicago Manual recommendations.