  We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless
symmetry-protected topological (gSPT) phase characterized by the coexistence of
the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom
and the symmetry-protected edge state due to spin degrees of freedom. This gSPT
phase has an interesting feature that no symmetry forbids the trivial spin gap
opening but a discrete symmetry, $\mathbb Z_3\times\mathbb Z_2^T$, forbids the
unique gapped ground state. In the first part of the paper, we numerically show
the coexistence of a critical entanglement entropy and a nontrivially
degenerate entanglement spectrum based on the density-matrix renormalization
group (DMRG) method.Next, we clarify that chirality degrees of freedom form the
Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane
state based on a degenerate perturbation theory. Last but not least, we discuss
the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local
$\mathbb{Z}_3$ rotation. We can thus characterize our gSPT phase as a
symmetry-protected critical phase protected by the $\mathbb{Z}_3$ on-site
symmetry, the $\mathbb{Z}_2^T$ time-reversal symmetry, the lattice translation
symmetry, and the U(1) spin-rotation symmetry.
