In quantum information, much can be done with the averaging formula

$$ \int (zz^*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{Proj}_{\mathrm{Sym}^t}. $$

Here the integral denotes an average over all unit vectors $z \in \mathbb{C}^n$ and $\operatorname{Proj}_{\mathrm{Sym}^t}$ denotes the orthogonal projector onto the totally symmetric subspace of $(\mathbb{C}^n)^{\otimes t}$. This result appears as Lemma 1 in the following paper.

Does this result have an analog for the average over all unit vectors $x \in \mathbb{R}^n$:

$$ \int (xx^\top)^{\otimes t} dx = {?} $$

I can verify by direct computations that

$$ \int (xx^\top)^{\otimes t} dz \ne {n + t -1 \choose t}^{-1} \operatorname{Proj}_{\mathrm{Sym}^t} \quad \text{for } t = 2. $$

Having a formula for $\int (xx^\top)^{\otimes t} dx$ (even for $t = 2$) would be very helpful for a problem I'm working on but I have not been able to find a reference for this or discover a formula myself.