Sounds very cool. Not impressed with the cross platform claim when the first thing I see is that android web is not supported, though.
I was hoping the underlying Android web issue would have been fixed by now (as I first noticed it ~3 years ago with some indication for progress), but looks like LiveStore needs a custom workaround for it. You can track the progress here: https://github.com/livestorejs/livestore/issues/321
I hope you understand that bridging the gaps between various levels of supported web APIs take a lot of effort and is non-trivial when building ambitious systems like LiveStore.
https://developer.mozilla.org/en-US/docs/Web/API/Web_Locks_A...
It is supported by all browsers. You just have all the tabs try to lock the same resource and have them return a promise in the lock callback. The first one wins and when that tab closes the next one in line gets automatically elected.
The leader can then use a broadcast channel and act as a server for all the other tabs to serialize access to any shared resources.