  We develop a method for describing the tropical complete intersection of a
tropical hypersurface and a tropical plane in $\mathbb{R}^3$. This involves a
method for determining the topological type of the intersection of a tropical
plane curve and $\mathbb{R}_{\leq 0}^2$ by using a polyhedral complex. As an
application, we study smooth tropical complete intersection curves of genus $3$
in $\mathbb{R}^3$. In particular, we show that there are no smooth tropical
complete intersection curves in $\mathbb{R}^3$ whose skeletons are the lollipop
graph of genus $3$. This gives a partial answer to a problem of Morrison.
