Magnetic fields permeate the diffuse interstellar medium (ISM) of the Milky Way, and are essential to explain the dynamical evolution and current shape of the Galaxy. Magnetic fields reveal themselves via their influence on the surrounding matter, and as such are notoriously hard to measure independently of other tracers. In this work, we attempt to disentangle an all sky map of the line-of-sight parallel component of the Galactic magnetic field from the Faraday effect, utilizing several tracers of the Galactic electron density. Additionally, we aim to produce a Galactic electron dispersion measure map and quantify several tracers of the structure of the ionized medium of the Milky Way. The method developed to reach these aims is based on information field theory, a Bayesian inference framework for fields, which performs well when handling noisy and incomplete data and constraining high dimensional parameter spaces. We rely on compiled catalogs of extragalactic Faraday rotation measures and Galactic pulsar dispersion measures, a well as data on bremsstrahlung and the hydrogen {alpha} spectral line to trace the ionized medium of the Milky Way. We present the first full sky map of the line-of-sight averaged Galactic magnetic field. Within this map, we find LoS parallel and LoS-averaged magnetic field strengths of up to 4uG, with an all-sky root-mean-square of 1.1uG, which is consistent with previous local measurements and global magnetic field models. Additionally, we produce a detailed electron dispersion measure map, which agrees with already existing parametric models at high latitudes, but suffers from systematic effects in the disk. Further analysis of our results with regard to the 3D structure of n_e_ reveals that it follows a Kolmogorov-type turbulence for most of the sky. From the reconstructed dispersion measure and emission measure maps we construct several tracers of variability of n_e_ along the LoS. This work demonstrates the power of consistent joint statistical analysis including multiple data sets and physical quantities and defines a roadmap towards a full three-dimensional joint reconstruction of the Galactic magnetic field and the ionized ISM.