Abstract Pine Island and Thwaites glaciers dominate Antarctica’s sea‐level contribution. However, it remains unclear whether their ice loss is driven by anomalously warm present‐day ocean conditions, perhaps linked to anthropogenic climate change, or are an ongoing response to historical natural climate anomalies. Here, we probe these drivers by subjecting three state‐of‐the‐art ice‐sheet models to an extreme hypothetical scenario of zero ocean melting. Pine Island responds by thickening and re‐advancing to the prominent seabed ridge on which it was grounded prior to 1940. In contrast, Thwaites continues to lose ice (at a decreasing rate) for 150 years, despite the absence of ocean melting. Since all present‐day ocean forcing via melt is removed, this demonstrates that the ongoing Thwaites ice loss is at least partly supported by the response to a historical perturbation. It also demonstrates that Thwaites’ present‐day mass loss trend cannot be reversed by ocean cooling alone, over policy‐relevant centennial timescales.