  The Haydys-Witten equations are partial differential equations on
five-dimensional Riemannian manifolds that are equipped with a non-vanishing
vector field $v$. Conjecturally, their solutions determine the Floer
differential in a gauge-theoretic approach to Khovanov homology. This article
introduces a certain decoupled version of the Haydys-Witten equations, a
specialization of the Haydys-Witten equations that exhibits a Hermitian
Yang-Mills structure. These equations exist whenever the vector bundle defined
by the orthogonal complement of $v$ admits an almost Hermitian structure. We
investigate the relation between the full Haydys-Witten equations and their
decoupled version on manifolds with poly-cylindrical ends and boundaries, and
find conditions under which the Haydys-Witten equations reduce to the decoupled
equations. This relies on a Weitzenb\"ock-like formula that shows that the
difference between the full Haydys-Witten equations and the decoupled equations
is governed by the asymptotic behaviour of solutions near boundaries and
non-compact ends. Regarding the analysis near boundaries, we provide a detailed
analysis of the polyhomogeneous expansion of Haydys-Witten solutions with
twisted Nahm pole boundary conditions, generalizing work of Siqi He in the
untwisted case. The corresponding analysis at non-compact ends relies on a
vanishing theorem by Nagy-Oliveira that was recently generalized by the author.
