  Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$,
the complete graph on ten vertices, contains a non-split three-component link;
that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of
Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known
intrinsically triple-linked graphs in $\mathbb{R}^3$ to include several other
families of graphs. In this paper, we will show that while some of these graphs
can be embedded 3-linklessly in $\mathbb{R}P^3$, $K_{10}$ is intrinsically
triple-linked in $\mathbb{R}P^3$.
