  We compute the exact path integral of $\mathcal{N}=2$ supersymmetric gauge
theories with general gauge group on $\mathbb{RP}^4$ and a
$\mathbb{Z}_2$-quotient of the hemi-$S^4$. By specializing to $SU(2)$
superconformal quivers, we show that these, together with hemi-$S^4$ partition
functions, compute Liouville correlators on unoriented/open Riemann surfaces.
We perform explicit checks for Riemann surfaces obtained as $\mathbb{Z}_2$
quotients of the sphere and the torus. We also discuss the coupled $3d\!-\!4d$
systems associated to Liouville amplitudes with boundary punctures.
