What does “would not have been possible had I remained” mean?

Conditions and Their Consequents

A conditional statement has two separate clauses, each with its own subject and verb: the condition itself, and the consequent that occurs only should that condition be met. There are fancy Greek words for these two pieces (protasis and apodosis), but I’ll stick to condition and consequent here.

The important thing for you to understand is that the ordering of the conditional’s two clauses doesn’t matter, because it is always clear which is the condition. You can have condition followed by consequent, or you can have consequent followed by condition — which is what’s going on with your particular example.

Additionally, clauses are also potentially subject to auxiliary inversion:

• For the condition, this inversion is mandatory when the conjunction (generally if or unless) is omitted under if-deletion. This is a special inversion rule.
• For the consequent, any inversion that happens occurs under normal inversion rules not special one as with the condition, like saying So will I or Neither will I instead of I will too.

Here follow several simple examples, saying which sort of production is operative in each in the parentheses following.

condition → consequent

• If you don’t go, I will. (condition → consequent)
• If you go, so will I. (condition → consequent-in-inversion)
• If I had anything better to do, I’d aleady be doing it. (condition → consequent)
• Had I anything better to do, I’d aleady be doing it. (condition-in-inversion → consequent)
• Unless you’ve something better to do, come with me. (condition → consequent)
• If you should have nothing better to do, you can come with me. (condition → consequent)
• Should you have nothing better to do, you can come with me. (condition-in-inversion → consequent)
• Had you anything better to do, you wouldn’t be here. (condition-in-inversion → consequent)

consequent ← condition

• I'll go if you do. (consequent ← condition)
• I'll go unless you do. (consequent ← condition)
• I’d already be doing it if I had anything better to do. (consequent ← condition)
• I’d already be doing it had I anything better to do. (consequent ← condition-in-inversion)
• You can come with me you if you’ve nothing better to do. (consequent ← condition)
• Come with me you unless you’ve something better to do. (consequent ← condition)
• You can come with me should you have nothing better to do. (consequent ← condition-in-inversion)

Rewriting Your Sentence

As the linked-to questions show, your example sentence is one of the lattermost variety: it places the consequent before the condition, and it uses if-deletion and inversion in that condition. Rewriting your sentence through successive transformations shows you exactly what it means:

1. I am sure this would not have been possible had I remained a typical Anglophone North American. (consequent ← condition-in-inversion)
2. I am sure this would not have been possible if  I had remained a typical Anglophone North American. (consequent ← condition)
3. Had I remained a typical Anglophone North American, I am sure this would not have been possible. (condition-in-inversion → consequent)
4. If  I had remained a typical Anglophone North American, I am sure this would not have been possible. (condition → consequent)

English has always supported bare conditionals like this, those without a conjunction. Back when it was more strongly inflected, this was always clear because that inversion also triggered a subjunctive inflection. Our last productive vestige of that is with the special form were.

• Were there any other way, I’d have tried it. (condition-in-inversion → consequent)
• Be it ever so humble, there’s no place like home. (condition-in-inversion → consequent)

Nowadays a bare conditional is most easily recognized by the obligatory inversion it demands.

1. I am sure this would not have been possible [had I remained a typical Anglophone North American].

Your original example has the same meaning as:

• 2) "I am sure this would not have been possible [if I had remained a typical Anglophone North American]."

Notice how the original version (#1) had subject-auxiliary inversion ("had I") for that subordinate clause "had I remained a typical Anglophone North American." That inversion has the semantic meaning of a conditional adjunct ("if I had") for your sentence.

Your #1 version has the same meaning as the #2 version which uses an overt conditional construction.

Another member had posted a link to a page with some related info: http://www.grammar-quizzes.com/10-8.html

If you need more info, go ahead and ask. :)