  We study unitarily equivalent bilateral weighted shifts with operator
weights. We establish a general characterization of unitary equivalence of such
shifts under the assumption that the weights are quasi-invertible. We prove
that under certain assumptions unitary equivalence of bilateral weighted shifts
with operator weights defined on $ \mathbb{C}^{2} $ can always be given by a
unitary operator with at most two non-zero diagonals. We provide examples of
unitarily equivalent shifts with weights defined on $ \mathbb{C}^{k} $ such
that every unitary operator, which intertwines them has at least $ k $ non-zero
diagonals.
