Install a trigger for your spreadsheet with change type "OTHER" and test for whether it runs when a new tab is added. I'm not sure if it works, but you can try it for yourself.
From code editor, choose "Edit" and "Current Triggers"
button - Add trigger
There is obviously a lot more that you'd need to do, but if you want it automated, then that's the starting point.
The spreadsheet can't find what browser tab your web app is in, and reload it. So, the Web App would need to "poll" for a change. I'm not sure if that is feasible for you.