  Understanding the origin of fast radio bursts (FRB's) is a central unsolved
problem in astrophysics that is severely hampered by their poorly determined
distance scale. Determining the redshift distribution of FRB's appears to
require arcsecond angular resolution, in order to associate FRB's with host
galaxies. In this paper, we forecast prospects for determining the redshift
distribution without host galaxy associations, by cross-correlating FRB's with
a galaxy catalog such as the SDSS photometric sample. The forecasts are
extremely promising: a survey such as CHIME/FRB that measures catalogs of $\sim
10^3$ FRB's with few-arcminute angular resolution can place strong constraints
on the FRB redshift distribution, by measuring the cross-correlation as a
function of galaxy redshift $z$ and FRB dispersion measure $D$. In addition,
propagation effects from free electron inhomogeneities modulate the observed
FRB number density, either by shifting FRB's between dispersion measure (DM)
bins or through DM-dependent selection effects. We show that these propagation
effects, coupled with the spatial clustering between galaxies and free
electrons, can produce FRB-galaxy correlations which are comparable to the
intrinsic clustering signal. Such effects can be disentangled based on their
angular and $(z, D)$ dependence, providing an opportunity to study not only
FRB's but the clustering of free electrons.
