  Let $O$ be chosen uniformly at random from the group of $(N+L) \times (N+L)$
orthogonal matrices. Denote by $\tilde{O}$ the upper-left $N \times N$ corner
of $O$, which we refer to as a truncation of $O$. In this paper we prove two
conjectures of Forrester, Ipsen and Kumar (2020) on the number of real
eigenvalues $N^{(m)}_{\mathbb{R}}$ of the product matrix $\tilde{O}_{1}\ldots
\tilde{O}_{m}$, where the matrices $\{\tilde{O}_{j}\}_{j=1}^{m}$ are
independent copies of $\tilde{O}$. When $L$ grows in proportion to $N$, we
prove that $$ \mathbb{E}(N^{(m)}_{\mathbb{R}}) = \sqrt{\frac{2m
L}{\pi}}\,\mathrm{arctanh}\left(\sqrt{\frac{N}{N+L}}\right) + O(1), \qquad N
\to \infty. $$ We also prove the conjectured form of the limiting real
eigenvalue distribution of the product matrix. Finally, we consider the
opposite regime where $L$ is fixed with respect to $N$, known as the regime of
weak non-orthogonality. In this case each matrix in the product is very close
to an orthogonal matrix. We show that $\mathbb{E}(N^{(m)}_{\mathbb{R}}) \sim
c_{L,m}\,\log(N)$ as $N \to \infty$ and compute the constant $c_{L,m}$
explicitly. These results generalise the known results in the one matrix case
due to Khoruzhenko, Sommers and \.{Z}yczkowski (2010).
