How do I read the event stream in JavaScript?

The browser's EventSource can't send Authorization headers, so read the stream with fetch on your server (Node 18+ has fetch built in):

const res = await fetch(`${BASE}/v1/runs/${runId}/events?after=${after}`, { headers });
const reader = res.body.pipeThrough(new TextDecoderStream()).getReader();
let buf = "";
for (;;) {
  const { value, done } = await reader.read();
  if (done) break;
  buf += value;
  let i;
  while ((i = buf.indexOf("\n\n")) >= 0) {
    const chunk = buf.slice(0, i); buf = buf.slice(i + 2);
    const event = /^event: (.*)$/m.exec(chunk)?.[1];
    const data = JSON.parse(/^data: (.*)$/m.exec(chunk)?.[1] ?? "null");
    // handle status / step / done / timeout / error
  }
}

The full reconnecting client is in the Streaming guide.