  Two asymptotic configurations on a full $\mathbb{Z}^d$-shift are
indistinguishable if for every finite pattern the associated sets of
occurrences in each configuration coincide up to a finitely supported
permutation of $\mathbb{Z}^d$. We prove that indistinguishable asymptotic pairs
satisfying a "flip condition" are characterized by their pattern complexity on
finite connected supports. Furthermore, we prove that uniformly recurrent
indistinguishable asymptotic pairs satisfying the flip condition are described
by codimension-one (dimension of the internal space) cut and project schemes,
which symbolically correspond to multidimensional Sturmian configurations.
Together the two results provide a generalization to $\mathbb{Z}^d$ of the
characterization of Sturmian sequences by their factor complexity $n+1$. Many
open questions are raised by the current work and are listed in the
introduction.
