The relativistic Hartree-Fock-Bogoliubov (RHFB) framework has been extended to incorporate axially symmetric octupole deformation, leading to the development of the O-RHFB model. This model is applied to analyze the onset of octupole deformation in ${}^{228}\mathrm{Th}$ and the proton-drip-line candidate ${}^{114}\mathrm{Ba}$ using the RHF Lagrangian PKA1 and the relativistic mean field (RMF) parameter set DD-ME2. Both interactions successfully reproduce stable octupole and quadruple deformations in the ground state of ${}^{228}\mathrm{Th}$. For ${}^{114}\mathrm{Ba}$, however, PKA1 yields a relatively soft octupole deformation, in contrast to the stable one predicted by DD-ME2. From the perspective of pseudospin symmetry (PSS), it is shown that the direct development of octupole deformation from a spherical configuration is unfavorable in both nuclei, and its onset is inevitably accompanied by significant quadruple deformation. Further analysis of PSS restoration in spherical configurations reveals a notable model discrepancy rooted in the description of nuclear binding. In particular, the exchange degrees of freedom associated with the $\pi$-pseudovactor and $\rho$-tensor couplings in PKA1 are found to play an essential role in determining nuclear binding and, consequently, in shaping the octupole-deformed characteristics of both ${}^{228}\mathrm{Th}$ and ${}^{114}\mathrm{Ba}$.

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• Nuclear shapes and moments
• Nuclear structure & decays
• Nucleon distribution
• Nuclear many-body theory