To find the subject, keep reading until an appropriate candidate comes up! Typically, the subject will be the first noun you see that isn't part of a modifier. Looking at the second independent clause (after the semicolon), here's how we might proceed:
at stake--This is a prepositional phrase. That's a modifier, so "stake" can't be the subject.
,should a significant number of these debts be repudiated,--This is a modifier. One clue is that it's set off with commas. Another is that it is serving as a qualifier. If this happens, the main action will happen, but we haven't seen what that main action is yet.
is--This is our verb, but where's the subject?
the solvency --Here's our subject! It's the first noun that is not part of a modifier.
of some of the world’s largest multinational banks.--This is a long modifying phrase. It simply adds details to the word "solvency."
So, shortening things down, the simple core of the clause is "At stake is the solvency." In more traditional order, that's "The solvency is at stake." The reversed Yoda-talk here is a turn of phrase equivalent to "What is at stake is . . . " One point of this usage (aside from making the problem harder) is to create a sense of drama: something is at stake. What is it? This construction also makes the very long modifier on the subject a bit less unwieldy by placing it at the end. Consider the two sentences below:
Not present at the meeting was my boss. There is no reason to write the sentence this way, unless I am contrasting with another, longer sentence "Present at the meeting were X, Y, and Z. Not present was my boss." As it is, it's just backwards!
Not present at the meeting were the many supporters of the plan, none of whom received invitations. Here, the subject has such a long modifier that it makes sense to move the whole thing to the end. The sentence wouldn't be
wrong the other way ("The many supporters of the plan, none of whom received invitations, were not present at the meeting.") However, it may not make its point as clearly and forcefully. Also, until we get to the last word, we don't know what the invitations are referring to.
For a similar construction, take a look at this one:
what-was-as-remarkable-as-the-development-of-the-compact-139154.html