  Let $P$ and $Q$ be finite point sets of the same cardinality in
$\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs
$G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if
for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with
the same clockwise ordering of the vertices on its boundary as in $f$. In
particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same
connected $n$-vertex graph.
  Deciding whether two labelled point sets admit compatible geometric paths is
known to be NP-complete. We give polynomial-time algorithms to find compatible
paths or report that none exist in three scenarios: $O(n)$ time for points in
convex position; $O(n^2)$ time for two simple polygons, where the paths are
restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for
points in general position if the paths are restricted to be monotone.
