While it isn't 100% war and conspiracy, Attack on Titan definitely has elements from both of those within it.
I highly recommend Attack on Titan if you like intense shows that don't bullshit you with "no one ever dies" and the hero is always there on time.
Hunter x Hunter is also recommended, though it is more similar to action anime with the whole mystic powers and stuff. It looks childish from the first episode (the 2011 reboot, at least as it's the only version I have watched), but it gets dark pretty quick.
Space Brothers is also good, but it's very different than most anime. It isn't filled with fan service, isn't quite drawn the way a lot of anime are, and is very much set in reality. Its main draw would be the character development rather than the story itself. It has many funny moments.
Gintama is hard to recommend to someone new to anime, but I do recommend it after watching some of the other bigger anime titles as it often references them. Gintama is a comedy anime with occasional serious action moments. Not for everyone and has a lot of dirty humor littered all over, so it really depends on your type of humor whether or not you'll like it.