Background: Theoretical calculations of the four-particle scattering above the four-cluster breakup threshold are technically very difficult due to nontrivial singularities or boundary conditions. Further complications arise when the long-range Coulomb force is present. Purpose: We aim at calculating proton-${}^3$He elastic scattering observables above three- and four-cluster breakup threshold. Methods: We employ Alt, Grassberger, and Sandhas (AGS) equations for the four-nucleon transition operators and solve them in the momentum-space framework using the complex-energy method whose accuracy and practical applicability is improved by a special integration method. Results: Using realistic nuclear interaction models we obtain fully converged results for the proton-${}^3$He elastic scattering. The differential cross section, proton and ${}^3$He analyzing powers, spin correlation and spin transfer coefficients are calculated at proton energies ranging from 7 to 35 MeV. Effective three- and four-nucleon forces are included via the explicit excitation of a nucleon to a $Delta$ isobar. Conclusions: Realistic proton-${}^3$He scattering calculations above the four-nucleon breakup threshold are feasible. There is quite good agreement between the theoretical predictions and experimental data for the proton-${}^3$He scattering in the considered energy regime. The most remarkable disagreements are the peak of the proton analyzing power at lower energies and the minimum of the differential cross section at higher energies. Inclusion of the $Delta$ isobar reduces the latter discrepancy.
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