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 zsim7 of Delta^2<181 kJy/sr * mK (95%) at ellsim800. 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--Lyalpha EOR cross spectrum.
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