  We determine the missing finite-size corrections to the asymptotic one-loop
dilatation operator of the real $\beta$-deformed $\mathcal{N}=4$ SYM theory for
the gauge groups $U(N)$ and $SU(N)$ in the 't Hooft limit. In the $SU(N)$ case,
the absence of the $U(1)$ field components leads to a new kind of finite-size
effect, which we call prewrapping. We classify which states are potentially
affected by prewrapping at generic loop orders and comment on the necessity to
include it into the integrability-based description. As a further result, we
identify classes of $n$-point correlation functions which at all loop orders in
the planar theory are given by the values of their undeformed counterparts.
Finally, we determine the superconformal multiplet structure and one-loop
anomalous dimensions of all single-trace states with classical scaling
dimension $\Delta_0 \leq 4.5$.
