This is an interesting theory problem, but it might be clearer to talk about shooters at a firing range, separated by partitions such that they can each only talk to their neighbors. The barriers eliminates the trivial solution of the shout: "Fire on the count of 3!" that common sense would provide.

The simplest solution would be to establish a clock, because with a clock you can say "Fire at 1pm exactly!". But even this is interesting because it requires shooters to know where they are in the line in order to take into account any signal propogation delay. If each person takes 1s to repeat a message that they hear, then it takes n seconds to go down the line, and each shooter must know how to compensate for that delay. E.g. the first shooter hears "it's noon", the next shooter hears "it's noon" but knows that he's 1s away from the source, and can calculate that the time is actually 12:01, the next shooter knows it's 12:02 etc. They could synchronize clocks, specify a time "shoot at 1pm!" and get the job done.

But what if we take away their clocks? This is where my intuition fails me, because I don't see how the problem can be solved without clocks. If you don't have clocks, you need to somehow get a signal to every shooter simultaneously, which I think is theoretically impossible, since without a clock the shooter cannot execute instructions on their own.

Which probably means I don't fully understand the scenario :)