  Cross-correlating 21cm and Ly$\alpha$ intensity maps of the Epoch of
Reionization (EoR) promises to be a powerful tool for exploring the properties
of the first galaxies. Next-generation intensity mapping experiments such as
the Hydrogen Epoch of Reionization Array (HERA) and SPHEREx will individually
probe reionization through the power spectra of the 21cm and Ly$\alpha$ lines
respectively, but will be limited by bright foregrounds and instrumental
systematics. Cross-correlating these measurements could reduce systematics,
potentially tightening constraints on the inferred astrophysical parameters. In
this study, we present forecasts of cross-correlation taking into account the
effects of exact uv-sampling and foreground filtering to estimate the
feasibility of HERAxSPHEREx making a detection of the 21cm-Ly$\alpha$
cross-power spectrum. We also project the sensitivity of a cross-power spectrum
between HERA and the proposed next-generation Cosmic Dawn Intensity Mapper. By
isolating the sources of uncertainty, we explore the impacts of experimental
limitations such as foreground filtering and Ly$\alpha$ thermal noise
uncertainty have on making a detection of the cross-power spectrum. We then
implement this strategy in a simulation of the cross-power spectrum and
observational error to identify redshifts where fiducial 21cmFAST models
predict the highest signal-to-noise detection ($z \sim 8$). We conclude that
detection of the SPHEREx-HERA cross-correlation will require an optimistic
level of 21cm foreground filtering, as well as deeper thermal noise
integrations due to a lack of overlapping sensitive modes but for CDIM with its
larger range of scales and lower noise forecast detection levels, may be
possible even with stricter 21cm foreground filtering.
