  Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with
Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique
to bound $\# X(K)$ when $\mathrm{rank}\ J(K) < g$. We study limitations of a
variant called `Restriction of Scalars Chabauty' (RoS Chabauty). RoS Chabauty
typically bounds $\# X(K)$ when $\mathrm{rank}\ J(K) \leq [K:\mathbb{Q}] (g -
1)$, but fails in the presence of a subgroup obstruction, a high-rank subgroup
scheme of $\mathrm{Res}_{K/\mathbb{Q}} J$ which intersects the image of
$\mathrm{Res}_{K/\mathbb{Q}} X$ in higher-than-expected dimension. We define
BCP obstructions, which are certain subgroup obstructions arising from the
geometry of $X$. BCP obstructions explain all known examples where RoS Chabauty
fails to bound $\# X(K)$. We also extend RoS Chabauty to compute $S$-integral
points on affine curves.
  Suppose $K$ does not contain a CM-subfield. We present a $p$-adic algorithm
which conjecturally computes solutions to the $S$-unit equation $x+y = 1$ for
$x,y \in \mathcal{O}_{K,S}^{\times}$ by using RoS Chabauty to compute
$S$-integral points on certain genus $0$ affine curves. As evidence the
algorithm succeeds, we prove that all but one of these curves have no subgroup
obstructions and that the remaining curve has no BCP obstructions to RoS
Chabauty. In contrast, under a generalized Leopoldt conjecture, we prove that
analogous methods using classical Chabauty cannot bound solutions to the
$S$-unit equation when $[K:\mathbb Q] \geq 3$ and $K$ is not totally real.
