The destabilization of climate sub-systems by one dominant positive feedback is thought to cause tipping points (TP) at a well-defined forcing parameter threshold. However, the coupling of sub-systems, competing feedbacks, and spatial heterogeneity may promote subtle, but abrupt reorganizations of geophysical flows before a catastrophic TP. Using a primitive-equation global ocean model we simulate a collapse of the Atlantic Meridional Overturning Circulation (AMOC) due to increasing glacial melt. Significantly prior to the collapse, various abrupt, qualitative changes in AMOC variability occur. These intermediate tipping points (ITP) are transitions between multiple stable circulation states. Using 2.4 million years of model simulations, we uncover a very rugged stability landscape featuring parameter regions of up to nine coexisting stable states. The path to an AMOC collapse via a sequence of ITPs depends on the rate of change of the meltwater input. This limits our ability to predict and define safe limits for TPs.