  Let $\mathfrak{i}$ denote the minimal cardinality of a maximal independent
family and let $\mathfrak{a}_T$ denote the minimal cardinality of a maximal
family of pairwise almost disjoint subtrees of $2^{<\omega}$. Using a countable
support iteration of proper, $^\omega\omega$-bounding posets of length
$\omega_2$ over a model of CH, we show that consistently
$\mathfrak{i}<\mathfrak{a}_T$. Moreover, we show that the inequality can be
witnessed by a co-analytic maximal independent family of size $\aleph_1$ in the
presence of a $\Delta^1_3$ definable well-order of the reals. The main result
of the paper can be viewed as a partial answer towards the well-known open
problem of the consistency of $\mathfrak{i}<\mathfrak{a}$.
