  We show that if a braid $B$ can be parametrised in a certain way, then
previous work can be extended to a construction of a polynomial
$f:\mathbb{R}^4\to\mathbb{R}^2$ with the closure of $B$ as the link of an
isolated singularity of $f$, showing that the closure of $B$ is real algebraic.
In particular, we prove that closures of squares of strictly homogeneous braids
and certain lemniscate links are real algebraic. We also show that the
constructed polynomials satisfy the strong Milnor condition, providing an
explicit fibration of the complement of the closure of $B$ over $S^1$.
