We resolve the computational complexity of two problems known as NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, showing that they are PPA-complete. We do this via a PPA-completeness result for an approximate version of the CONSENSUS-HALVING problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision. At the heart of our construction is a smooth embedding of the high-dimensional M\"{o}bius strip in the CONSENSUS-HALVING problem. These results settle the status of PPA as a class that captures the complexity of "natural'" problems whose definitions do not incorporate a circuit.
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