ringing in the matrix ${}_+ I_{(lm)(lm)'}$. (For the weight function adopted here, the width of the main peak of ${}_\pm I_{(lm)(lm)'}$ is given by the inverse of the survey size, but the effective band-limit is larger being set by the inverse size of the apodized region.)

To emphasise the importance of $E$-$B$ mixing in the structure of the covariance matrix for $\tilde{C}_l^B$, in Fig. \[fig:exactBB\_galcosine\] we show both the exact $\rmn{cov}(\tilde{C}_l^B,\tilde{C}_{l'}^B)$ and the matrix we obtain by setting $C_l^E=0$. The latter is very inaccurate on the diagonal at $l \la 120 $, where $E$-$B$ mixing is most important, and particularly so at the largest scales where it further fails to pick up the additional power in $C_l^E$ from reionization. Setting $C_l^E$ to zero also artificially suppresses the correlations: there is significantly more structure off the diagonal in the left-hand plot in Fig. \[fig:exactBB\_galcosine\] than on the right. This reflects the fact that ${}_-I_{(lm)(lm)'}$ is intrinsically broader than ${}_+ I_{(lm)(lm')}$ (see Fig. \[fig:power\_and\_window\]).

The covariance matrix $\rmn{cov}(\tilde{C}_l^E,\tilde{C}_{l'}^B)$, shown in Fig. \[fig:exactEB\_galcosine\], is only non-zero because of $E$-$B$ mixing. It is correspondingly broader than $\rmn{cov}(\tilde{C}_l^E,\tilde{C}_{l'}^E)$. The dominant contribution is from $E$-mode power, so the covariance matrix peaks near the diagonal at the 