I get the feeling it's path-dependent. If I start uptown and you start downtown the algorithm could potentially show us two different cities. So we would have to agree on a city beforehand.
Edit: depending on how complex the world is, the server could probably just send out the world changes itself, rather than relying on each client to correctly (deterministically) apply the changes. It needs to do so anyhow for late joiners.
You can partially work around this by requiring the client to use a server-provided PRNG seed, although even then a kind of aim bot could help the player decide which path to explore to get favourable tiles.
So your best bet to avoid cheating is to do the collapse on the server using a hidden RNG.
of course tiles would need to be large enough to keep latency reasonable.