Zero-knowledge proofs are mathematical cryptographic methods to demonstrate the validity of a claim while providing no further information beyond the claim itself. The possibility of using such proofs to process classified and other sensitive physical data has attracted attention, especially in the field of nuclear arms control. Here we demonstrate a non-electronic fast neutron differential radiography technique using superheated emulsion detectors that can confirm that two objects are identical without revealing their geometry or composition. Such a technique could form the basis of a verification system that could confirm the authenticity of nuclear weapons without sharing any secret design information. More broadly, by demonstrating a physical zero-knowledge proof that can compare physical properties of objects, this experiment opens the door to developing other such secure proof-systems for other applications.