  We use full sky simulations, including the effects of foreground
contamination and removal, to explore multi-tracer synergies between a SKA-like
21cm intensity mapping survey and a LSST-like photometric galaxy redshift
survey. In particular we study ratios of auto and cross-correlations between
the two tracers as estimators of the ratio of their biases, a quantity that
should benefit considerably from the cosmic variance cancellation of the
multi-tracer approach. We show how well we should be able to measure the bias
ratio on very large scales (down to $\ell \sim 3$), which is crucial to measure
primordial non-Gaussianity and general relativistic effects on large scale
structure. We find that, in the absence of foregrounds but with realistic noise
levels of such surveys, the multi-tracer estimators are able to improve on the
sensitivity of a cosmic-variance contaminated measurement by a factor of $2-4$.
When foregrounds are included, estimators using the 21cm auto-correlation
become biased. However, we show that cross-correlation estimators are immune to
this and do not incur in any significant penalty in terms of sensitivity from
discarding the auto-correlation data. However, the loss of long-wavelength
radial modes caused by foreground removal in combination with the low redshift
resolution of photometric surveys, reduces the sensitivity of the multi-tracer
estimator, albeit still better than the cosmic variance contaminated scenario
even in the noise free case. Finally we explore different alternative avenues
to avoid this problem.
