Proper classes of maximal θ-independent families from large cardinals

While maximal independent families can be constructed from ZFC via Zorn's lemma, the presence of a maximal σ-independent family already gives an inner model with a measurable cardinal, and Kunen has shown that from a measurable cardinal one can construct a forcing extension in which there is a maximal σ-independent family. We extend this technique to construct proper classes of maximal θ-independent families for various uncountable θ. In the first instance, a single θ⁺-strongly compact cardinal has a set-generic extension with a proper class of maximal θ-independent families. In the second, we take a class-generic extension of a model with a proper class of measurable cardinals to obtain a proper class of θ for which there is a maximal θ-independent family.