$\mathrm{Re}(E\{\mathbf{x}\mathbf{y}^H\} )=\mathbf {0}\Leftrightarrow E\{\mathbf{x}^H\mathbf{y}\} = 0$?

Let $\mathbf{x}$ and $\mathbf{y}$ two complex (column) random vector of dimension $m \times 1$. Is it true that:\begin{equation}\mathrm{Re}(E\{\mathbf{x}\mathbf{y}^H\} ) = \mathbf{0}\Leftrightarrow E\{\mathbf{x}^H\mathbf{y}\} = 0?\end{equation}