To calculate the angular momentum of a body we need to specify a point (or an axis?) from which to define the displacement vector $\vec{r}$, so that $\vec{L} = \vec{r} \times \vec{p}$.
For a rigid body, the formula becomes $\vec{L}=I\vec{\omega}$, assuming the moment of inertia is not a tensor. So in this case we need to specify an axis.
NOW: what if I wanted to calculate the total angular momentum of the Earth-Moon system? What is the most sensible point/axis to choose?
I would say, intuitively, the centre of mass of the system.
So about the CoM,
$$\begin{align}\vec{L}_\text{tot} &= \vec{L}\text{ of the Earth due to its orbit about centre of mass} \\ &+ \vec{L}\text{ of Moon due to its orbit about centre of mass} \\ &+ \text{angular momenta due to rotations of Earth and Moon around their own axes}?\end{align}$$
I am not sure if and how to include the rotations of Moon and Earth around their axes: I know they have to enter somehow because of the tidal friction effect on the orbit of the Moon, but I don't know how to reconcile this with the fact that I chose the centre of mass as the point of reference here, and none of the rotation axes have anything to do with the centre of mass of the Earth-Moon system.