Constraining the equation of state (EOS) of strongly interacting, dense matter is the focus of intense experimental, observational, and theoretical effort. Chiral effective field theory ($\chi$EFT) can describe the EOS between the typical densities of nuclei and those in the outer cores of neutron stars while perturbative QCD (pQCD) can be applied to properties of deconfined quark matter, both with quantified theoretical uncertainties. However, describing the complete range of densities between nuclear saturation and an almost-free quark gas with a single EOS that has well-quantified uncertainties is a challenging problem. In this work, we argue that Bayesian multi-model inference from χEFT and pQCD can help bridge the gap between the two theories: we combine the Gaussian random variables that constitute the theories’ predictions for the pressure as a function of the density in symmetric nuclear matter. We do this using two Bayesian model mixing procedures: a pointwise approach, and a correlated approach implemented via a Gaussian process (GP), and present results for the pressure and speed of sound in each. The second method produces a smooth $\chi$EFT-to-pQCD EOS. Without input data in the intermediate region, the choice of prior on the EOS, encoded through the GP kernel, as the prior on the EOS function space significantly affects the result in that region. We also discuss future extensions and applications to neutron star matter guided by recent EOS constraints from nuclear theory, nuclear experiment, and multi-messenger astronomy.