<p>Quantum error correction is essential for mitigating hardware errors in quantum computers by encoding logical information into several physical qubits. However, no single error-correcting code intrinsically supports a fault-tolerant implementation of all the gates needed for universal quantum computing. One approach for addressing this problem is to switch between two suitable error-correcting codes that in combination provide a fault-tolerant universal gate set. Here we present the experimental implementation of fault-tolerant code switching between two different codes in a trapped-ion processor. We switch between the 7-qubit colour code, which features fault-tolerant CNOT and <i>H</i> quantum gates, and the 10-qubit code, which allows for a fault-tolerant <i>T</i> gate implementation. Together, these codes form a complementary universal gate set. We construct logical circuits and prepare 12 different logical states that are not accessible natively in a fault-tolerant way within a single code. Finally, we use code switching to entangle two logical qubits using the full universal gate set in a single logical quantum circuit. Our results experimentally demonstrate a route towards deterministic control over logical qubits with low auxiliary qubit overhead and without relying on the probabilistic preparation of resource states.</p>