  A detection of the predicted anticorrelation between 21cm and either Ly-alpha
or H-alpha from the Epoch of Reionization (EOR) would be a powerful probe of
the first galaxies. While 3D intensity maps isolate foregrounds in low
k_\parallel modes, infrared surveys cannot yet match the field of view and
redshift resolution of radio intensity mapping experiments. In contrast, 2D
(i.e., broad band) infrared intensity maps can be measured with current
experiments and are limited by foregrounds instead of photon or thermal noise.
We show 2D experiments can measure most of the 3D fluctuation power at k<0.2
Mpc^-1 while preserving its correlation properties. However, we show
foregrounds pose two challenges: (1) simple geometric effects produce
percent-level correlations between radio and infrared fluxes, even if their
luminosities are uncorrelated; and (2) radio and infrared foreground residuals
contribute sample variance noise to the cross spectrum. The first challenge
demands better foreground masking and subtraction, while the second demands
large fields of view to average away uncorrelated radio and infrared power.
Using radio observations from the Murchison Widefield Array and near-infrared
observations from the Asteroid Terrestrial-impact Last Alert System, we set an
upper limit on residual foregrounds of the 21cm--Ly-alpha cross power spectrum
at z\sim7 of \Delta^2<181 kJy/sr * mK (95\%) at \ell\sim800. We predict levels
of foreground correlation and sample variance noise in future experiments,
showing that higher resolution surveys such as LOFAR, SKA-LOW, and the Dark
Energy Survey can start to probe models of the 21cm--Ly\alpha EOR cross
spectrum.
