Let me ask a variant of this question but for the ternary Cantor set. We have two continuous complex-valued functions on the Cantor set $\Delta$; call them $f,g$. For $\varepsilon >0$, can we find another pair of such functions $h,k$ such that $fg = hk$, $\|f-h\|_\infty, \|g-k\|_\infty<\varepsilon$ (the supremum norm), and $|h(t)|+|k(t)|>0$ for all $t$?
That'd be the case if $\{f=0\} \cap \{g = 0\}$ were clopen.