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We present the first numerical simulations that track the evolution of ablack hole-neutron star (BH-NS) merger from pre-merger to$r\gtrsim10^{11}\,{\rm cm}$. The disk that forms after a merger of mass ratio$q=2$ ejects massive disk winds ($3-5\times10^{-2}\,M_{\odot}$). We introducevarious post-merger magnetic configurations and find that initial poloidalfields lead to jet launching shortly after the merger. The jet maintains aconstant power due to the constancy of the large-scale BH magnetic flux untilthe disk becomes magnetically arrested (MAD), where the jet power falls off as$L_j\sim t^{-2}$. All jets inevitably exhibit either excessive luminosity dueto rapid MAD activation when the accretion rate is high or excessive durationdue to delayed MAD activation compared to typical short gamma-ray bursts(sGRBs). This provides a natural explanation for long sGRBs such as GRB 211211Abut also raises a fundamental challenge to our understanding of jet formationin binary mergers. One possible implication is the necessity of higher binarymass ratios or moderate BH spins to launch typical sGRB jets. For post-mergerdisks with a toroidal magnetic field, dynamo processes delay jet launching suchthat the jets break out of the disk winds after several seconds. We show forthe first time that sGRB jets with initial magnetization $\sigma_0>100$ retainsignificant magnetization ($\sigma\gg1$) at $r>10^{10}\,{\rm cm}$, emphasizingthe importance of magnetic processes in the prompt emission. The jet-windinteraction leads to a power-law angular energy distribution by inflating anenergetic cocoon whose emission is studied in a companion paper.

Large-scale Evolution of Seconds-long Relativistic Jets from Black Hole-Neutron Star Mergers