  A vector is \emph{dyadic} if each of its entries is a dyadic rational number,
i.e. of the form $\frac{a}{2^k}$ for some integers $a,k$ with $k\geq 0$. A
linear system $Ax\leq b$ with integral data is \emph{totally dual dyadic} if
whenever $\min\{b^\top y:A^\top y=w,y\geq {\bf 0}\}$ for $w$ integral, has an
optimal solution, it has a dyadic optimal solution. In this paper, we study
total dual dyadicness, and give a co-NP characterization of it in terms of
\emph{dyadic generating sets for cones and subspaces}, the former being the
dyadic analogue of \emph{Hilbert bases}, and the latter a polynomial-time
recognizable relaxation of the former. Along the way, we see some surprising
turn of events when compared to total dual integrality, primarily led by the
\emph{density} of the dyadic rationals. Our study ultimately leads to a better
understanding of total dual integrality and polyhedral integrality. We see
examples from dyadic matrices, $T$-joins, cycles, and perfect matchings of a
graph.
