The goal of haplotype-aware genome assembly is to reconstruct all individual haplotypes from a mixed sample and to provide corresponding abundance estimates. We provide a reference-genome-independent solution based on the construction of a variation graph, capturing all diversity present in the sample. We solve the contig abundance estimation problem and propose a greedy algorithm to efficiently build maximal-length haplotypes. Finally, we obtain accurate frequency estimates for the reconstructed haplotypes through linear programming techniques. Our method outperforms state-of-the-art approaches on viral quasispecies benchmarks and has the potential to assemble bacterial genomes in a strain-aware manner as well.