  Measurement-free quantum error correction (MFQEC) offers an alternative to
standard measurement-based QEC in platforms with an unconditional qubit reset
gate. We revisit the question of fault tolerance (FT) for a measurement-free
variant of the Steane code that leverages multi-qubit gates and redundant
syndrome extraction, finding previously overlooked phase-flip errors that
undermine FT. We then construct a revised MFQEC circuit that is resistant to
all single-qubit errors, but which nonetheless cannot tolerate certain
correlated errors. In order to investigate FT systematically, we introduce an
efficient method to classically simulate MFQEC circuits with (i) Clifford gates
for syndrome extraction, (ii) syndrome-controlled Pauli operations for
decoding, and (iii) a Pauli noise model. We thereby find a pseudothreshold of
$\sim0.7\%$ for our revised MFQEC Steane code under a restricted noise model
previously considered in the literature. We then relax noise model assumptions
to identify general requirements for FT with multi-qubit gates, finding that
existing multi-qubit neutral atom gates are incompatible with fault-tolerant
syndrome extraction in a straightforward implementation of both
measurement-based and measurement-free variants of the Steane code. Decomposing
multi-qubit gates into two-qubit gates similarly spoils FT. Finally, we discuss
the theoretical ingredients that are necessary to recover FT for MFQEC codes,
including single-shot FT and a recent proposal by Heu{\ss}en \textit{et
al.}~[arXiv:2307.13296] to achieve FT by ``copying'' errors onto an ancilla
register. By combining multi-qubit gates, redundant syndrome extraction, and
copy-assisted FT, we construct a measurement-free and fault-tolerant variant of
the Steane code with a pseudothreshold of $\sim0.1\%$.
