  Quantum state teleportation is commonly used in designs for large-scale
fault-tolerant quantum computers. Using Quantinuum's H2 trapped-ion quantum
processor, we implement the first demonstration of a fault-tolerant state
teleportation circuit for a quantum error correction code - in particular, the
planar topological [[7,1,3]] color code, or Steane code. The circuits use up to
30 trapped ions at the physical layer qubits and employ real-time quantum error
correction - decoding mid-circuit measurement of syndromes and implementing
corrections during the protocol. We conduct experiments on several variations
of logical teleportation circuits using both transversal gates and lattice
surgery protocols. Among the many measurements we report on, we measure the
logical process fidelity of the transversal teleportation circuit to be
0.975(2) and the logical process fidelity of the lattice surgery teleportation
circuit to be 0.851(9). Additionally, we run a teleportation circuit that is
equivalent to Knill-style quantum error correction and measure the process
fidelity to be 0.989(2).
