  We construct an infinite tower of covering spaces over the configuration
space of $n-1$ distinct non-zero points in the complex plane. This results in
an action of the braid group $\mathbb{B}_n$ on the set of $n$-adic integers
$\mathbb{Z}_n$ for all natural numbers $n\geq 2$. We study some of the
properties of these actions such as continuity and transitivity. The
construction of the actions involves a new way of associating to any braid $B$
an infinite sequence of braids, whose braid types are invariants of $B$. We
present computations for the cases of $n=2$ and $n=3$ and use these to show
that an infinite family of braids close to real algebraic links, i.e., links of
isolated singularities of real polynomials $\mathbb{R}^4\to\mathbb{R}^2$.
