So what is the syntax to pass Requires-External to setuptools/distutils?

There is none by default, as neither distutils nor setuptools support the field. Also, requires_external keyword arg is not supported as well - it is silently ignored, just as any other unknown keyword arg.

To add local support for requires_external kwarg, you need to provide your own distribution implementation. Minimal example (requires Python 3.9):

from setuptools.dist import Distribution as DistributionOrig

class Distribution(DistributionOrig):
    _DISTUTILS_UNSUPPORTED_METADATA = DistributionOrig._DISTUTILS_UNSUPPORTED_METADATA | {'requires_external': dict}

You can now pass requires_external to setup() when paired with distclass:

setup(
    ...,
    requires_external=[
        'C',
        'libpng (>=1.5)',
        'make; sys_platform != "win32"',
    ],
    distclass=Distribution,
)

The next goal is to actually write the contents of requires_external to PKG-INFO. This is quite tricky to do via distribution metadata itself because setuptools patches the relevant methods (DistributionMetadata.read_pkg_file() and DistributionMetadata.write_pkg_file()) with own monolithic impls. The easiest way for this IMO is to modify PKG-INFO post factum via a custom egg_info impl:

import email
from pathlib import Path
from setuptools.command.egg_info import egg_info as egg_info_orig


class egg_info(egg_info_orig):
    def run(self):
        super().run()
        # PKG-INFO is now guaranteed to exist
        pkg_info_file = Path(self.egg_info, 'PKG-INFO')
        pkg_info = email.message_from_bytes(pkg_info_file.read_bytes())
        for req in self.distribution.metadata.requires_external:
            pkg_info.add_header('Requires-External', req)
        pkg_info_file.write_bytes(pkg_info.as_bytes())

Pass your own egg_info impl via cmdclass argument in setup():

setup(
    ...,
    requires_external=[
        'C',
        'libpng (>=1.5)',
        'make; sys_platform != "win32"',
    ],
    distclass=Distribution,
    cmdclass={'egg_info': egg_info},
)