How can I synchronously determine a JavaScript Promise's state?

No such synchronous inspection API exists for native JavaScript promises. It is impossible to do this with native promises. The specification does not specify such a method.

Userland libraries can do this, and if you're targeting a specific engine (like v8) and have access to platform code (that is, you can write code in core) then you can use specific tools (like private symbols) to achieve this. That's super specific though and not in userland.

Nope, no sync API, but here's my version of the async promiseState (with help from @Matthijs):

function promiseState(p) {
  const t = {};
  return Promise.race([p, t])
    .then(v => (v === t)? "pending" : "fulfilled", () => "rejected");
}

var a = Promise.resolve();
var b = Promise.reject();
var c = new Promise(() => {});

promiseState(a).then(state => console.log(state)); // fulfilled
promiseState(b).then(state => console.log(state)); // rejected
promiseState(c).then(state => console.log(state)); // pending

promise-status-async does the trick. It is async but it does not use then to wait the promise to be resolved.

const {promiseStatus} = require('promise-status-async');
// ...
if (await promiseStatus(promise) === 'pending') {
    const idle = new Promise(function(resolve) {
        // can do some IDLE job meanwhile
    });
    return idle;
}