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We introduce a new, general-purpose, range-separated hybrid van der Waals density \ph{functional, termed vdW-DF-ahbr,} within the non-empirical vdW-DF method [JPCM 32, 393001 (2020)]. It combines correlation from vdW-DF2 with a screened Fock exchange that is fixed by \ph{a new model of exchange effects} in the density-explicit vdW-DF2-b86r functional [PRB 89, 121103(R) (2014)]. The new vdW-DF2-ahbr prevents spurious exchange binding and has a small-density-gradient form set from many-body perturbation analysis. It is accurate for \ph{bulk as well as layered materials} and it systematically and significantly improves the performance of present vdW-DFs for molecular problems. Importantly, vdW-DF2-ahbr also outperforms present-standard (dispersion-corrected) range-separated hybrids on a broad collection of noncovalent-interaction benchmark sets, while at the same time successfully mitigating the density-driven errors that often affect the description of molecular transition states and isomerization calculations. vdW-DF2-ahbr furthermore improves on state of the art density functional theory approaches by 1) correctly predicting both the substrate structure and the site preference for CO adsorption on Pt(111), 2) outperforming existing non-empirical vdW-DFs for the description of CO$_2$ adsorption in both a functionalized and in a simple metal-organic framework, and 3) being highly accurate \ph{for the} set of base-pair interactions in a model of DNA assembly.