We prove various results that, given a sufficiently rich subgroup $G$ of the group of homeomorphisms on the real line, describe the structure of the other possible actions of $G$ on the real line, and address under which conditions such actions must be semi-conjugate to the natural defining action of $G$. The main assumption is that $G$ should be locally moving, meaning that for every open interval the subgroup of elements whose support is contained in such interval acts on it without fixed points. One example (among many others) is given by Thompson's group $F$. A first rigidity result implies that if $G$ is a locally moving group, every faithful minimal action of $G$ on the real line by $C^1$ diffeomorphisms is conjugate to its standard action. It turns out that the situation is much wilder when considering actions by homeomorphisms: for a large class of groups, including Thompson's group $F$, we describe uncountably many conjugacy classes of minimal faithful actions by homeomorphisms on the real line. To gain insight on such exotic actions, we introduce and develop the notion of $\mathbb{R}$-focal action, a class of actions on the real line that can be encoded by certain actions by homeomorphisms on planar real trees fixing an end. Under a suitable finite generation condition on a locally moving group $G$, we prove that every minimal faithful action of $G$ on the line is either conjugate to the standard action, or it is $\mathbb{R}$-focal and the action on the associated real tree factors via a horofunction onto the standard action of $G$ on the line. Among applications of this result, we show that for a large class of locally moving groups, the standard action is locally rigid, in the sense that every sufficiently small perturbation in the compact-open topology gives a semi-conjugate action. This is based on an analysis of the space of harmonic actions on the line for such groups.