  Let $\mathcal{R}$ be an expansion of the ordered real additive group. When
$\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an
ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval
$I\subseteq \mathbb{R}$, or $\mathcal{R}$ is a reduct of an ordered vector
space. We say $\mathcal{R}$ is field-type if it satisfies the former condition.
In this paper, we prove a more general result for arbitrary expansions of
$(\mathbb{R},<,+)$. In particular, we show that for expansions that do not
define dense $\omega$-orders (we call these type A expansions), an appropriate
version of Zilber's principle holds. Among other things we conclude that in a
type A expansion that is not field-type, every continuous definable function
$[0,1]^m \to \mathbb{R}^n$ is locally affine outside a nowhere dense set.
