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Background: Theoretical calculations of the four-particle scattering abovethe four-cluster breakup threshold are technically very difficult due tonontrivial singularities or boundary conditions. Further complications arisewhen the long-range Coulomb force is present.  Purpose: We aim at calculating proton-${}^3$He elastic scattering observablesabove three- and four-cluster breakup threshold.  Methods: We employ Alt, Grassberger, and Sandhas (AGS) equations for thefour-nucleon transition operators and solve them in the momentum-spaceframework using the complex-energy method whose accuracy and practicalapplicability is improved by a special integration method.  Results: Using realistic nuclear interaction models we obtain fully convergedresults for the proton-${}^3$He elastic scattering. The differential crosssection, proton and ${}^3$He analyzing powers, spin correlation and spintransfer coefficients are calculated at proton energies ranging from 7 to 35MeV. Effective three- and four-nucleon forces are included via the explicitexcitation of a nucleon to a $\Delta$ isobar.  Conclusions: Realistic proton-${}^3$He scattering calculations above thefour-nucleon breakup threshold are feasible. There is quite good agreementbetween the theoretical predictions and experimental data for theproton-${}^3$He scattering in the considered energy regime. The most remarkabledisagreements are the peak of the proton analyzing power at lower energies andthe minimum of the differential cross section at higher energies. Inclusion ofthe $\Delta$ isobar reduces the latter discrepancy.

Calculation of proton-${}^3$He elastic scattering between 7 and 35 MeV