  We study deformed Hermitian Yang-Mills (dHYM) connections on ruled surfaces
explicitly, using the momentum construction. As a main application we provide
many new examples of dHYM connections coupled to a variable background K\"ahler
metric. These are solutions of the moment map partial differential equations
given by the Hamiltonian action of the extended gauge group, coupling the dHYM
equation to the scalar curvature of the background. The large radius limit of
these coupled equations is the K\"ahler-Yang-Mills system of
\'Alvarez-C\'onsul, Garcia-Fernandez and Garc\'ia-Prada, and in this limit our
solutions converge smoothly to those constructed by Keller and
T{\o}nnesen-Friedman. We also discuss other aspects of our examples including
conical singularities, realisation as B-branes, the small radius limit and
canonical representatives of complexified K\"ahler classes.
