fetch-mock has a function patchNativeFetchForSafari that wraps the native fetch when it detects it's running in Safari. By a funny coincidence, what fetch-mock looks for is exactly how jsdom identifies itself.

The side effect is that, when jsdom-global is loaded, the patchNativeFetchForSafari function only passes the first argument on to the nativeFetch function. Like so:

require('jsdom-global/register');
const fetchMock = require('fetch-mock');

fetchMock.config.fetch = function() {
  console.log(arguments);
  return Promise.reject(new Error('nyeh'));
}

const sandbox = fetchMock.sandbox();
sandbox.spy(true);

sandbox('http://localhost:2021', { headers: { 'cookie': 'id=5' } }).catch(() => {});

Produces:

fluggo@NONEYABIZ:~/software/fetchmocktest$ node test.js 
[Arguments] { '0': 'http://localhost:2021/' }

...leaving out the specified header.

Either the detection needs to be more specific, nativeFetch should be called as nativeFetch.apply(arguments), or both. For my purposes, my workaround will probably be navigator = { vendor: 'Compy' };.