If you play it during the Requisition phase, can the other SF player put a Tech on a ship, have the Baseleios remove it with Elaborate Muntions (EM), and then just play the same Tech again?
There doesn't seem to be a downside to EM otherwise. Maybe there wasn't supposed to be, but it still seems kinda... odd.
If you are just using EM to gain a single W however, I think you are not getting full value from the card. EM is best used on Techs which either give immediate Install bonuses (like, say, Baseleios' Tendral Atomizer), or on W/D based Techs that are party damaged so that come back at full strength.
Also, there seems to be a lot of confusion here (and in your other thread question) on when Boost cards can be played. So let me take a moment to clarify this. On the back of the rulebook you will see all the sub-phases that take place during each main phase. For instace, the Requisition phase starts with Icon Activation, then to credit counting, then to purchasing. All those phases happen simultaneously when they occur and you may not play Boost cards in the middle of that proccess! Only between those phases can you play Boosts (it's a super common mistake).
So for the Installation portion of the [I] phase, you can't Install a tech, play Elaborate Munitions, then re-install. You have to play the Boost either before or after the entire Install portion happens.
Except page 13 says, "Boost - Installation" card can only be played during the Installation Phase." It doesn't say before or after it, but "during" it. Elaborate Munitions isn't Boost - Installation, but the implication seems to be that generic Boost cards can also be played during a phase.
That seems to contradict the previous paragraph, admittedly. And also, during the... Phase isn't the same as during an Event. But the "during" wording is a tad confusing. Which is probably why it's a common mistake.