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We compute the exact path integral of $\mathcal{N}=2$ supersymmetric gaugetheories 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-$S^4$ partitionfunctions, 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.

$\mathcal{N}=2$ gauge theories on unoriented/open four-manifolds and their AGT counterparts