Base editors (BEs) enable efficient, programmable installation of point mutations while avoiding the use of double-strand breaks. Simultaneous application of two or more different BEs, such as an adenine BE (which converts A·T base pairs to G·C) and a cytosine BE (which converts C·G base pairs to T·A), is not feasible because guide RNA crosstalk results in non-orthogonal editing, with all BEs modifying all target loci. Here we engineer both adenine BEs and cytosine BEs that can be orthogonally multiplexed by using RNA aptamer-coat protein systems to recruit the DNA-modifying enzymes directly to the guide RNAs. We generate four multiplexed orthogonal BE systems that enable rates of precise co-occurring edits of up to 7.1% in the same DNA strand without enrichment or selection strategies. The addition of a fluorescent enrichment strategy increases co-occurring edit rates up to 24.8% in human cells. These systems are compatible with expanded protospacer adjacent motif and high-fidelity Cas9 variants, function well in multiple cell types, have equivalent or reduced off-target propensities compared with their parental systems and can model disease-relevant point mutation combinations.