What are some career options for Math PhDs who want to work in non-Western countries (e.g. Southeast Asia, Central Asia, Africa)?
This answer is mostly for Asia.
Yes, there are programs in at least Japan and Korea (that I am aware of) for American Ph.D to find a job, including professorship.

I think the largest difference/challenge you will face is that many times, being a student and working in Asian countries such as China, Japan, Korea, is very different.  In general, the faculty, students, local businesses, etc. are happy to accommodate students spending some time in their region, learning their language, and living.  Systems are in place for this and the universities most often actively encourage it.

Meetings

As a professor it can be (but not always) different.  Unless you learn the language fluently before you go, think about your new colleagues and how they will feel, being required to speak to you in a language other than their own.  Not necessarily about the weather, but more importantly (what I noticed) is in faculty meetings, budget decisions, etc.  In this case, you can easily be the only person not able to understand, which leaves two options: you are left out of the meetings or everyone else is forced to speak your language (if they can).  

You mention not caring about prestige, but I would caution you that many 'bad' universities have a lower chance of faculty being comfortable in English.

Administration

Beyond faculty, which may or may not speak English, you still will have administration to deal with.  Do you assume your contract will be in English? While some online university systems are in English, I have seen many parts that are not, such as the email system in English, but the login systems or vacation registration is not.

Culture

Of course, you need to think about culture.  The idea of Asian culture may allure you, but remember it is not always what it seems (just as people going to America find out).  In at least Japan and Korea, Impact Factor is the largest measure of performance, and you must follow it.  Be ready to change your work style if it does not fit into the way you are evaluated.

Resources

You can check https://www.higheredjobs.com/international/  for international job postings.