  A new class of groups $\mathcal{C}$, containing all coherent RAAGs and all
toral relatively hyperbolic groups, is defined. It is shown that, for a group
$G$ in the class $\mathcal{C}$, the $\mathbb{Z}[t]$-exponential group
$G^{\mathbb{Z}[t]}$ may be constructed as an iterated centraliser extension.
Using this fact, it is proved that $G^{\mathbb{Z}[t]}$ is fully residually $G$
(i.e. it has the same universal theory as $G$) and so its finitely generated
subgroups are limit groups over $G$. If $\mathbb{G}$ is a coherent RAAG, then
the converse also holds - any limit group over $\mathbb{G}$ embeds into
$\mathbb{G}^{\mathbb{Z}[t]}$. Moreover, it is proved that limit groups over
$\mathbb{G}$ are finitely presented, coherent and CAT$(0)$, so in particular
have solvable word and conjugacy problems.
