Tom Clancy's H.A.W.X. 2

Select 'Playstation Network' from the main menu and then Player Match. Use to filter the results so that Co-op Story mode is selected, and join an existing session, or alternatively press to host your own. Either way, you must play Co-op Story and NOT Co-op Mission in order for this trophy to pop. I know this because I played Co-op Mission and it did not pop, I also tried hosting my own and it still didn't. However it awarded me the trophy immediately after completing a mission in Co-op Story. It does not matter which story mission you actually complete.

I might also add, I joined the Co-op Story session with around 3 bad guys left; the existing players killed them, and the mission ended without me having killed anyone, and being present for about 20 seconds. The trophy still popped straight away.