  We study the birational complexity of log Calabi-Yau $3$-folds. For such a
pair $(X,B)$ of index one and coregularity zero, we show that $c_{\rm
bir}(X,B)\in \{0,2,3\}$. Further, we prove that $(X,B)$ has a log Calabi-Yau
crepant birational model that admits a crepant contraction to
$(\mathbb{P}^{3-c},H_0+\dots+H_{3-c})$, where $c=c_{\rm bir}(X,B)$. To prove
this, we give a geometric characterization of standard $\mathbb{P}^1$-links.
