Summary: ``Spin precession is one of the key physical effects that coul unveil the origin of the compact binaries detected by ground- and space-based gravitational-wave (GW) detectors, and shed light on their possible formation channels. Efficiently and accurately modeling the GW signals emitted by these systems is crucial to extract their properties. Here, we present {\tt SEOBNRv5PHM}, a multipolar precessing-spin waveform model within the effective-one-body formalism for the full signal (i.e. inspiral, merger and ringdown) of binary black holes (BBHs). In the nonprecessing limit, the model reduces to {\tt SEOBNRv5HM}, which is calibrated to 442 numerical-relativity (NR) simulations, 13 waveforms from BH perturbation theory, and nonspinning energy flux from second-order gravitational self-force theory. We remark that {\tt SEOBNRv5PHM} is not calibrated to precessing-spin NR waveforms from the Simulating eXtreme Spacetimes Collaboration. We validate {\tt SEOBNRv5PHM} by computing the unfaithfulness against 1543 precessing-spin NR waveforms, and find that for 99.8\% (84.4\%) of the cases, the maximum value, in the total mass range $20-300M_\odot$, is below 3\% (1\%). These numbers reduce to 95.3\% (60.8\%) when using the previous version of the {\tt SEOBNR} family, {\tt SEOBNRv4PHM}, and to 78.2\% (38.3\%) when using the state-of-the-art frequency-domain multipolar precessing-spin phenomenological {\tt IMRPhenomXPHM} model. Due to much better computational efficiency of {\tt SEOBNRv5PHM} compared to {\tt SEOBNRv4PHM}, we are also able to perform extensive Bayesian parameter estimation on synthetic signals and GW events observed by LIGO-Virgo detectors. We show that {\tt SEOBNRv5PHM} can be used as a standard tool for inference analyses to extract astrophysical and cosmological information of large catalogs of BBHs.''MRPhenomXPHM