  The unconventional electronic ground state of Sr$_3$IrRuO$_7$ is explored via
resonant x-ray scattering techniques and angle-resolved photoemission
measurements. As the Ru content approaches $x=0.5$ in
Sr$_3$(Ir$_{1-x}$Ru$_x$)$_2$O$_7$, intermediate to the $J_{eff}=1/2$ Mott state
in Sr$_3$Ir$_2$O$_7$ and the quantum critical metal in Sr$_3$Ru$_2$O$_7$, a
thermodynamically distinct metallic state emerges. The electronic structure of
this intermediate phase lacks coherent quasiparticles, and charge transport
exhibits a linear temperature dependence over a wide range of temperatures.
Spin dynamics associated with the long-range antiferromagnetism of this phase
show nearly local, overdamped magnetic excitations and an anomalously large
energy scale of 200 meV---an energy far in excess of exchange energies present
within either the Sr$_3$Ir$_2$O$_7$ or Sr$_3$Ru$_2$O$_7$ solid-solution
endpoints. Overdamped quasiparticle dynamics driven by strong spin-charge
coupling are proposed to explain the incoherent spectral features of the
strange metal state in Sr$_3$IrRuO$_7$.
