If $B^{sing}=B^{pre-sing}$ is of codimension at least two in $B$, then there is a continuous path $\gamma(t)$ in the space of ${\bf Z}$-affine structures with singularities which connects a given structure with the one coming from a PL compactification, and such that for all $t\in[0,1]$ we have $codim(B_{t}^{pre-sing})\geq 2$ and $\gamma(t)$ has Finiteness and Independence properties.