If lieutennant Sai coordinates a coordinate action to a reaper, does he gets to do the coordinate again?

Or the other way around if the reaper coordinates Sai, which coordinates back...what's happening there? Can Sai make a new coordination as his own post-movment action?

Rules say: If an ability instructs a ship to coordinate, this is different than performing
a coordinate action. A ship that coordinates without performing the action can
still perform the coordinate action this round

21 hours ago, dmors said:

If lieutennant Sai coordinates a coordinate action to a reaper, does he gets to do the coordinate again?

No. Lt Sai did a coordinate action to the Reaper if the Reaper then decides to chain coordinate it would proc Sai's ability BUT he is still bound by the rule that a ship can only do a given action once per round. Since he already did a coordinate action he can't do it again.

21 hours ago, dmors said:

Or the other way around if the reaper coordinates Sai, which coordinates back...what's happening there? Can Sai make a new coordination as his own post-movment action?

Okay so this is a bit tangled but let me answer the second part first. No, for the same reason as above. There's a blanket rule that a ship may not perform the same action twice in a round. If Sai Coordinates back to the Reaper he can't coordinate again later in the same round.

The first part is a bit more interesting. If the Reaper goes first and coordinates to Sai, Sai could simply coordinate back to the Reaper if he wants. The process of coordiamating is choose a ship at range 0-2 and then the chosen ship performs an action. The receiving ship picks any action it likes, it isn't dictated by the coordinating ship. When Sai coordinates back to the Reaper the Reaper would simply pick any action other than coordinate (because it already did that this round) and then perform that action. If you only wanted the Reaper to get an action this would be a lot of unnecessary steps, but if the Reaper were to choose either Focus or Jam that would proc Sai and he could copy it as those both appear on his action bar as well. Let's say the Reaper does a Focus. Sai could Focus as well, then when his turn comes up he would move and then do any action other than Coordinate or Focus, again because he already did both of those this round.

21 hours ago, dmors said:

Rules say: If an ability instructs a ship to coordinate, this is different than performing a coordinate action. A ship that coordinates without performing the action can
still perform the coordinate action this round

This is worth keeping in mind but it is irrelevant to your specific questions here as nothing here "instructs a ship to coordinate". As I mentioned above when you coordinate to a ship you are not instructing the ship to do X. You are merely giving the ship an opportunity to perform an action. The chosen ship then performs an action of its choice. In gameplay or discussion people will shorthand the phrasing as like "I'll have Sai coordinate a Focus to the Reaper" for ease but mechanically whats actually happening is Sai coordinates with the chosen ship and then the ship performs one action of its choice from its available actions. So even if you "coordinate a coordinate" to someone they are still performing a coordinate action and thus locking it out for the rest of the round.

@sharrrp 's writeup is thorough and accurate. So, I'll just give the short, tl;dr version:

No. Because they are all coordinate actions , rather than instructions to coordinate. Lt. Sai takes a coordinate action and targets the Reaper. Reaper also chooses a coordinate action (giving it to whomever, doesn't matter). Then Lt. Sai is given the opportunity to take a coordinate action, because of his ability, but cannot because he's already taken that action this turn.