  We extend the pure pseudo-power-spectrum formalism proposed recently in the
context of the Cosmic Microwave Background polarized power spectra estimation
by Smith (2006) to incorporate cross-spectra computed for multiple maps of the
same sky area. We present an implementation of such a technique, paying
particular attention to a calculation of the relevant window functions and
mixing (mode-coupling) matrices. We discuss the relevance and treatment of the
residual $E/B$ leakage for a number of considered sky apodizations as well as
compromises and assumptions involved in an optimization of the resulting power
spectrum uncertainty. In particular, we investigate the importance of a
pixelization scheme, patch geometry, and sky signal priors used in apodization
optimization procedures. In addition, we also present results derived for more
realistic sky scans as motivated by the proposed balloon borne experiment EBEX.
We conclude that the presented formalism thanks to its speed and efficiency can
provide an interesting alternative to the CMB polarized power spectra
estimators based on the optimal methods at least on angular scales smaller than
~10 degrees. In this regime, we find that it is capable of suppressing the
total variance of the estimated $B$-mode spectrum to within a factor of ~2 of
the variance due to only the sampling and noise uncertainty of the B-modes
alone, as derived from the Fisher matrix approach.
