bgstaal/multipleWindow3dScene ? reverse-engineered prompt

I want to build a cool web experiment. Can you create a webpage that displays a simple 3D scene using Three.js? A few basic shapes and some simple lighting would be great to start.

The most important feature is that this 3D scene needs to be synchronized across multiple browser windows. If I have the page open and then I open the same page again in a new window, they should both be showing the same 3D world. When I rotate the camera or an object in one window, the other window should update in real-time to show the exact same change. It should feel like having two different views into a single, shared 3D space.

Please build this with plain JavaScript. I think you might need to use the browser's local storage to keep the state of the scene shared between the windows.