The Poynting vector (in Gaussian units) is given by $\mathbf{S} = \frac{c}{8\pi}\left(\mathbf{E}\times\mathbf{H}^*\right)$, and the magnetic field is given by $\mathbf{H} = \frac{c}{\mu\omega}\left(\mathbf{k}\times\mathbf{E}\right)$ where $\mathbf{k}$ is the propagation vector.
The average energy flux through a surface of normal unit vector $\mathbf{\hat{n}}$ is $$\Re\left\{\mathbf{S}\cdot\mathbf{\hat{n}}\right\} = \frac{c^2}{8\pi\omega\mu}\Re\left\{\mathbf{\hat{n}}\cdot\left[\mathbf{E}\times\left(\mathbf{k}^*\times\mathbf{E}^*\right)\right]\right\}$$
Using the vector triple product expansion I get: $$ = \frac{c^2}{8\pi\omega\mu}\Re\left\{\left(\mathbf{\hat{n}}\cdot\mathbf{k}^*\right)\left|\mathbf{E}\right|^2 - \left(\mathbf{\hat{n}}\cdot\mathbf{E}^*\right)\left(\mathbf{E}\cdot\mathbf{k}^*\right)\right\}$$
Now for an evanescent wave, the propagation vector $\mathbf{k}$ has a purely imaginary normal component, so $$\Re\left\{\left(\mathbf{\hat{n}}\cdot\mathbf{k}^*\right)\left|\mathbf{E}\right|^2\right\} = 0$$
Therefore I am left with: $$\Re\left\{\mathbf{S}\cdot\mathbf{\hat{n}}\right\} = -\frac{c^2}{8\pi\omega\mu}\Re\left\{\left(\mathbf{\hat{n}}\cdot\mathbf{E}^*\right)\left(\mathbf{E}\cdot\mathbf{k}^*\right)\right\}$$
In theory the evanescent wave is supposed to have zero energy flux across the surface, so this expression should equal zero. But I can't see this mathematically.
I know that $\mathbf{E}\cdot\mathbf{k} = 0$, but here we have $\mathbf{k}^*$ instead. We can use the relation $\mathbf{k}^* = \mathbf{k} - 2i\Im\left\{\mathbf{k}\right\}$ to write it as:
$$\Re\left\{\mathbf{S}\cdot\mathbf{\hat{n}}\right\} = \frac{c^2}{4\pi\omega\mu}\Re\left\{i\left(\mathbf{\hat{n}}\cdot\mathbf{E}^*\right)\left(\mathbf{E}\cdot\Im\left\{\mathbf{k}\right\}\right)\right\}$$
So this would be zero if $\left(\mathbf{\hat{n}}\cdot\mathbf{E}^*\right)\left(\mathbf{E}\cdot\Im\left\{\mathbf{k}\right\}\right)$ is purely real. But this does not look obvious to me, given that $\mathbf{E}$ is in general complex-valued as well (its components might have phase shifts, for example).
How can I show that the average of the normal Poynting vector is zero for an evanescent wave?