We show that finite-memory effects fundamentally reshape cascade risk in coupled climate tipping systems by decoupling ensemble stability from pathwise instability. Applying a regime-switching Volterra model with tempered fractional kernels to CMIP6 multi-model ensembles (<em>n</em> = 10 for the Atlantic Meridional Overturning Circulation, AMOC; <em>n</em> = 37 for the Amazon and Greenland), we demonstrate that three tipping elements operate under structurally distinct memory regimes linked to different physical processes. AMOC lower-tail occupancy triples under SSP5-8.5 while ensemble-mean weakening reaches only ≈ 0.5<em>σ</em>; a per-model-consistent memory amplification index <em>M</em><span style="position: relative; top: -.6em; left: -.55em;">^</span>≈ 2.7–6.0 confirms that persistence, not mean shift, is the primary driver. The Amazon presents a mechanistically contrasting picture (<em>M</em><span style="position: relative; top: -.6em; left: -.55em;">^</span>< 1): its tail amplification is forcing- dominated, making ensemble-mean drying projections adequate for risk assessment. Greenland internal surface-mass-balance (SMB) variability is strongly long-range dependent (<span style="text-decoration: overline;"><em>H</em></span> = 0.89; 89 % of models), anchoring it as a persistent upstream driver. Cascade simulations show that quenched (99th-percentile) pathwise Amazon damage exceeds annealed (median) projections by a factor of > 2 under weak forcing – a divergence invisible to ensemble summaries and absent in memory-free dynamics. These results demonstrate that neglecting long-range dependence systematically understates upper-tail cascade risk, and that AMOC, the Amazon, and Greenland require mechanistically differentiated treatment in climate-risk assessment.