  The combination of multiple cosmological probes can produce measurements of
cosmological parameters much more stringent than those possible with any
individual probe. We examine the combination of two highly correlated probes of
late-time structure growth: (i) weak gravitational lensing from a survey with
photometric redshifts and (ii) galaxy clustering and redshift space distortions
from a survey with spectroscopic redshifts. We choose generic survey designs so
that our results are applicable to a range of current and future photometric
redshift (e.g. KiDS, DES, HSC, Euclid) and spectroscopic redshift (e.g. DESI,
4MOST, Sumire) surveys. Combining the surveys greatly improves their power to
measure both dark energy and modified gravity. An independent, non-overlapping
combination sees a dark energy figure of merit more than 4 times larger than
that produced by either survey alone. The powerful synergies between the
surveys are strongest for modified gravity, where their constraints are
orthogonal, producing a non-overlapping joint figure of merit nearly 2 orders
of magnitude larger than either alone. Our projected angular power spectrum
formalism makes it easy to model the cross-correlation observable when the
surveys overlap on the sky, producing a joint data vector and full covariance
matrix. We calculate a same-sky improvement factor, from the inclusion of these
cross-correlations, relative to non-overlapping surveys. We find nearly a
factor of 4 for dark energy and more than a factor of 2 for modified gravity.
The exact forecast figures of merit and same-sky benefits can be radically
affected by a range of forecasts assumption, which we explore methodically in a
sensitivity analysis. We show that that our fiducial assumptions produce robust
results which give a good average picture of the science return from combining
photometric and spectroscopic surveys.
