  We discuss consistent truncations of eleven-dimensional supergravity on a
six-dimensional manifold $M$, preserving minimal $\mathcal{N}=2$ supersymmetry
in five dimensions. These are based on $G_S \subseteq USp(6)$ structures for
the generalised $E_{6(6)}$ tangent bundle on $M$, such that the intrinsic
torsion is a constant $G_S$ singlet. We spell out the algorithm defining the
full bosonic truncation ansatz and then apply this formalism to consistent
truncations that contain warped AdS$_5 \times_{\rm w}M$ solutions arising from
M5-branes wrapped on a Riemann surface. The generalised $U(1)$ structure
associated with the $\mathcal{N}=2$ solution of Maldacena-Nu\~nez leads to
five-dimensional supergravity with four vector multiplets, one hypermultiplet
and $SO(3)\times U(1)\times \mathbb{R}$ gauge group. The generalised structure
associated with "BBBW" solutions yields two vector multiplets, one
hypermultiplet and an abelian gauging. We argue that these are the most general
consistent truncations on such backgrounds.
