  Motivated in part by Propp's intruded Aztec diamond regions, we consider
hexagonal regions out of which two horizontal chains of triangular holes
(called ferns) are removed, so that the chains are at the same height, and are
attached to the boundary. By contrast with the intruded Aztec diamonds (whose
number of domino tilings contain some large prime factors in their
factorization), the number of lozenge tilings of our doubly-intruded hexagons
turns out to be given by simple product formulas in which all factors are
linear in the parameters. We present in fact $q$-versions of these formulas,
which enumerate the corresponding plane-partitions-like structures by their
volume. We also pose some natural statistical mechanics questions suggested by
our set-up, which should be possible to tackle using our formulas.
