In quantum information, much can be done with the averaging formula
$$ \int_{\mathbb{C}P^{n-1}} (zz^*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{\Pi}_{\mathrm{Sym}^t}$$
Here the integral gives an average over all unit vectors $z \in \mathbb{C}^n$ and $\operatorname{\Pi}_{\mathrm{Sym}^t}$ denotes the orthogonal projector onto the totally symmetric subspace of $(\mathbb{C}^n)^{\otimes t}$. This result appears as Lemma 1 in the following paperthe following paper.
Does this result have an analog for the average over all unit vectors $x \in \mathbb{R}^n$?
$$ \int_{\mathbb{R}P^{n-1}} (xx^\top)^{\otimes t} dx = {?} $$
By direct computations, I see that the first formula does not hold when $\mathbb{C}P$ is replaced by $\mathbb{R}P$ for $t=2$.
Having a formula for $\int_{\mathbb{R}P^{n-1}} (xx^\top)^{\otimes t} dx$ (even for $t = 2$) would help with a problem I'm working on, but I have not been able to find a reference for this or discover a formula myself.