Those turrets are the striped black and white turrets that need to be shot down twice. They come in 2 sizes but in my opinion, both count towards the achievement.
I found out that depending on your character, they spawn on rather different stages. I played mostly with Mitsuhide (Vasara 2's character with 3 stars in strength and only 1 in speed) and with him, those turrets only spawned after waves 5/6. However, if you use different characters, they will sometimes spawn as early as in wave 1.