  We sample the posterior probability distributions of the low-energy constants
(LECs) in $\Delta$-full chiral effective field theory ($\chi$EFT) up to third
order. We use eigenvector continuation for fast and accurate emulation of the
likelihood and Hamiltonian Monte Carlo to draw effectively independent samples
from the posteriors. Our Bayesian inference is conditioned on the Granada
database of neutron-proton ($np$) cross sections and polarizations. We use
priors grounded in $\chi$EFT assumptions and a Roy-Steiner analysis of
pion-nucleon scattering data. We model correlated EFT truncation errors using a
two-feature Gaussian process, and find correlation lengths for $np$ scattering
energies and angles in the ranges 45--83 MeV and 24--39 degrees, respectively.
These correlations yield a non-diagonal covariance matrix and reduce the number
of independent scattering data with a factor of 8 and 4 at the second and third
chiral orders, respectively. The relatively small difference between the second
and third order predictions in $\Delta$-full $\chi$EFT suppresses the marginal
variance of the truncation error and the effects of its correlation structure.
Our results are particularly important for analyzing the predictive
capabilities in \textit{ab initio} nuclear theory.
