  The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the
plane only aperiodically. They are the smallest known set of Wang tiles to do
so and are not as well understood as other examples of aperiodic Wang tiles. We
show that the $\mathbb{Z}^2$ action by translation on a certain subset of the
Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian
sequences (rotation sequences), is minimal. We give a characterization of this
space as a skew product as well as explicit bounds on the waiting time between
occurrences of $m\times n$ configurations.
