The soft X-ray transient EP241021a: A cosmic explosion with a complex off-axis jet and cocoon from a massive progenitor
X-ray flashes (XRFs) are fast X-ray transients thought to be softer analogs of gamma-ray bursts (GRBs). With its soft X-ray sensitivity, the Einstein Probe (EP) provides a unique opportunity to study these events. We report multiwavelength observations of EP241021a, a soft X-ray transient detected by EP, and interpret its afterglow in the context of leading XRF models. The prompt emission was observed by EP-WXT and Fermi-GBM, followed by a broad campaign across radio (uGMRT, ATCA, e-MERLIN, ALMA), optical (LBT, GTC, CAHA), and X-rays (EP-FXT). Light curves and spectra were analyzed with both empirical and physical models of GRBs and spherical expansions (both nonrelativistic and mildly relativistic cocoons). The afterglow shows multiple components, consistent with a structured jet interacting with a complex environment. The early optical and X-ray decline is explained by wide, low-Lorentz-factor ($\gamma \sim 40$) wings, while a rebrightening at approximately 7 days arises from the off-axis jet core. Radio data require an additional mildly relativistic cocoon ($\gamma \sim 2$), and a late (70 days) spectral component peaking at 50 GHz suggests a second, slower cocoon ($\gamma \sim 1$).