  We study $\mathrm{Spin}(7)$-manifolds with an effective multi-Hamiltonian
action of a four-torus. On an open dense set, we provide a Gibbons-Hawking type
ansatz that describes such geometries in terms of a symmetric $4\times4$-matrix
of functions. This description leads to the first known
$\mathrm{Spin}(7)$-manifolds with a rank $4$ symmetry group and full holonomy.
We also show that the multi-moment map exhibits the full orbit space
topologically as a smooth four-manifold, containing a trivalent graph in
$\mathbb{R}^4$ as the image of the set of the special orbits.
