For example, how would this hypothetical situation play out?
Imagine that Alice and Bob are aboard spaceships moving inertially with a relative speed of 0.8c. At some point they pass right next to each other, and Alice defines the position and time of their passing to be at position x = 0, time t = 0 in her frame, while Bob defines it to be at position x' = 0 and time t' = 0 in his frame. In Alice's frame she remains at rest at position x = 0, while Bob is moving in the positive x direction at 0.8c; in Bob's frame he remains at rest at position x' = 0, and Alice is moving in the negative x' direction at 0.8c. Each one also has a tachyon transmitter aboard their ship, which sends out signals that move at 2.4c in the ship's own frame.
When Alice's clock shows that 300 days have elapsed since she passed next to Bob (t = 300 days in her frame), she uses the tachyon transmitter to send a message to Bob, saying "Ugh, I just ate some bad shrimp". At t = 450 days in Alice's frame, she calculates that since the tachyon signal has been traveling away from her at 2.4c for 150 days, it should now be at position x = (2.4)*(150) = 360 light-days in her frame, and since Bob has been traveling away from her at 0.8c for 450 days, he should now be at position x = (0.8)*(450) = 360 light-days in her frame as well, meaning that this is the moment the signal catches up with Bob. So, in her frame Bob receives Alice's message at x = 360, t = 450. Due to the effects of time dilation, in her frame Bob is aging more slowly than she is by a factor of \frac{1}{ \gamma} = \sqrt{1 - { (v/c)^2}}, in this case 0.6, so Bob's clock only shows that 0.6*450 = 270 days have elapsed when he receives the message, meaning that in his frame he receives it at x' = 0, t' = 270.
When Bob receives Alice's message, he immediately uses his own tachyon transmitter to send a message back to Alice saying "Don't eat the shrimp!" 135 days later in his frame, at t' = 270 + 135 = 405, he calculates that since the tachyon signal has been traveling away from him at 2.4c in the -x' direction for 135 days, it should now be at position x' = -(2.4)*(135) = -324 light-days in his frame, and since Alice has been traveling at 0.8c in the -x direction for 405 days, she should now be at position x' = -(0.8)*(405) = -324 light-days as well. So, in his frame Alice receives his reply at x' = -324, t' = 405. Time dilation for inertial observers is symmetrical, so in Bob's frame Alice is aging more slowly than he is, by the same factor of 0.6, so Alice's clock should only show that 0.6*405 = 243 days have elapsed when she receives his reply. This means that she receives a message from Bob saying "Don't eat the shrimp!" only 243 days after she passed Bob, while she wasn't supposed to send the message saying "Ugh, I just ate some bad shrimp" until 300 days elapsed since she passed Bob, so Bob's reply constitutes a warning about her own future.
