  (abridged) Observations of Faraday rotation for extragalactic sources probe
magnetic fields both inside and outside the Milky Way. Building on our earlier
estimate of the Galactic contribution, we set out to estimate the extragalactic
contributions. We discuss the problems involved; in particular, we point out
that taking the difference between the observed values and the Galactic
foreground reconstruction is not a good estimate for the extragalactic
contributions. We point out a degeneracy between the contributions to the
observed values due to extragalactic magnetic fields and observational noise
and comment on the dangers of over-interpreting an estimate without taking into
account its uncertainty information. To overcome these difficulties, we develop
an extended reconstruction algorithm based on the assumption that the
observational uncertainties are accurately described for a subset of the data,
which can overcome the degeneracy with the extragalactic contributions. We
present a probabilistic derivation of the algorithm and demonstrate its
performance using a simulation, yielding a high quality reconstruction of the
Galactic Faraday rotation foreground, a precise estimate of the typical
extragalactic contribution, and a well-defined probabilistic description of the
extragalactic contribution for each data point. We then apply this
reconstruction technique to a catalog of Faraday rotation observations. We vary
our assumptions about the data, showing that the dispersion of extragalactic
contributions to observed Faraday depths is most likely lower than 7 rad/m^2,
in agreement with earlier results, and that the extragalactic contribution to
an individual data point is poorly constrained by the data in most cases.
