Chemically tagging groups of stars born in the same birth cluster is a major goal of spectroscopic surveys. To investigate the feasibility of such strong chemical tagging, we perform a blind chemical tagging experiment on abundances measured from APOGEE survey spectra. We apply a density-based clustering algorithm to the 8D chemical space defined by [Mg/Fe], [Al/Fe], [Si/Fe], [K/Fe], [Ti/Fe], [Mn/Fe], [Fe/H], and [Ni/Fe], abundances ratios which together span multiple nucleosynthetic channels. In a high-quality sample of 182 538 giant stars, we detect 21 candidate clusters with more than 15 members. Our candidate clusters are more chemically homogeneous than a population of non-member stars with similar [Mg/Fe] and [Fe/H], even in abundances not used for tagging. Group members are consistent with having the same age and fall along a single stellar-population track in log g versus Teff space. Each group's members are distributed over multiple kpc, and the spread in their radial and azimuthal actions increases with age. We qualitatively reproduce this increase using N-body simulations of cluster dissolution in Galactic potentials that include transient winding spiral arms. Observing our candidate birth clusters with high-resolution spectroscopy in other wavebands to investigate their chemical homogeneity in other nucleosynthetic groups will be essential to confirming the efficacy of strong chemical tagging. Our initially spatially compact but now widely dispersed candidate clusters will provide novel limits on chemical evolution and orbital diffusion in the Galactic disc, and constraints on star formation in loosely bound groups.