We present a new joint analysis of pulsar dispersion measures and diffuse H-alpha emission in the Milky Way, which we use to derive the density, pressure and filling factor of the thick disk component of the warm ionised medium (WIM) as a function of height above the Galactic disk. By excluding sightlines at low Galactic latitude that are contaminated by HII regions and spiral arms, we find that the exponential scale-height of free electrons in the diffuse WIM is 1830 (+120, -250) pc, a factor of two larger than has been derived in previous studies. The corresponding inconsistent scale heights for dispersion measure and emission measure imply that the vertical profiles of mass and pressure in the WIM are decoupled, and that the filling factor of WIM clouds is a geometric response to the competing environmental influences of thermal and non-thermal processes. Extrapolating the properties of the thick-disk WIM to mid-plane, we infer a volume-averaged electron density 0.014 +- 0.001 cm^-3, produced by clouds of typical electron density 0.34 +- 0.06 cm^-3 with a volume filling factor 0.04 +- 0.01. As one moves off the plane, the filling factor increases to a maximum of ~30% at a height of approximately 1-1.5 kpc, before then declining to accommodate the increasing presence of hot, coronal gas. Since models for the WIM with a ~1 kpc scale-height have been widely used to estimate distances to radio pulsars, our revised parameters suggest that the distances to many high-latitude pulsars have been substantially underestimated.
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