  Perverse schobers are categorical analogs of perverse sheaves. Examples arise
from varieties admitting flops, determined by diagrams of derived categories of
coherent sheaves associated to the flop: in this paper we construct mirror
partners to such schobers, determined by diagrams of Fukaya categories with
stops, for examples in dimensions 2 and 3. Interpreting these schobers as
supported on loci in mirror moduli spaces, we prove homological mirror symmetry
equivalences between them. Our construction uses the coherent-constructible
correspondence and a recent result of Ganatra-Pardon-Shende to relate the
schobers to certain categories of constructible sheaves. As an application, we
obtain new mirror symmetry proofs for singular varieties associated to our
examples, by evaluating the categorified cohomology operators of
Bondal-Kapranov-Schechtman on our mirror schobers.
