The ice streams flowing into the Amundsen Sea, West Antarctica, are losing mass due to changes in the oceanic basal melting of their floating ice shelves. Rapid ice-shelf melting is sustained by the delivery of warm Circumpolar Deep Water to the ice-shelf cavities, which is first supplied to the continental shelf by an undercurrent that flows eastward along the shelf break. Temporal variability of this undercurrent controls ice-shelf basal melt variability. Recent work shows that on decadal timescales the undercurrent variability opposes surface wind variability. Using a regional model, we show that undercurrent variability is driven by sea-ice freshwater fluxes, particularly those north of the shelf break, which affect the cross-shelf break density gradient. This sea-ice variability is caused by tropical Pacific variability impacting atmospheric conditions over the Amundsen Sea. Ice-shelf melting also feeds back onto the undercurrent by affecting the on-shelf density, thereby influencing shelf-break density gradient anomalies.