Many types of combinatorial, algebraic or measure-theoretic families of reals, such as mad families, Hamel bases or Vitali sets, can be framed as maximal independent sets in analytic hypergraphs on Polish spaces. Their existence is guaranteed by the Axiom of Choice, but low-projective witnesses ($\mathbf{Delta}^1_2$) were only known to exist

in general in models of the form $L[a]$ for a real $a$. Our main result is that, after a countable support iteration of Sacks forcing or for example splitting forcing (a less known forcing adding splitting reals) over L, every analytic hypergraph on a Polish space has a $\mathbf{\Delta}^1_2$ maximal independent set. As a corollary, this solves an open problem of Brendle, Fischer and Khomskii by providing a model with a $\Pi^1_1$ mif (maximal independent family) while the independence number $\mathfrak{i}$ is bigger than $\aleph_1$.