Accurately reconstructing the reticulate histories of polyploids remains a central challenge for understanding plant evolution. Although phylogenetic networks can provide insights into relationships among polyploid lineages, inferring networks may be hampered by the complexities of homology determination in polyploid taxa. We use simulations to show that phasing alleles from allopolyploid individuals can improve inference of phylogenetic networks under the multispecies coalescent. Phased allelic data can also improve divergence time estimates for networks, which is helpful for evaluating allopolyploid speciation hypotheses and proposing mechanisms of speciation. To achieve these outcomes, we present a novel pipeline that leverages a recently developed phasing algorithm to reliably phase alleles from polyploids. This pipeline is especially appropriate for target enrichment data, where depth of coverage is typically high enough to phase entire loci. We provide an empirical example in the North American Dryopteris fern complex that demonstrates how phasing can help reveal the mode of polyploidization and improve network inference. We establish that our pipeline (PATÉ: Phased Alleles from Target Enrichment data) is capable of recovering a high proportion of phased loci from both diploids and polyploids, and that these data improve network estimates compared to using haplotype consensus assemblies. This approach is shown to be especially effective in reticulate complexes where there are multiple hybridization events. The pipeline is available at: https://github.com/gtiley/Phasing.