  A new inner product is constructed on each standard module over the
Temperley-Lieb algebra $\mathsf{TL}_n(\beta)$ for $\beta\in \mathbb R$ and $n
\ge 2$. On these modules, the Hamiltonian $h = -\sum_i e_i$ is shown to be
self-adjoint with respect to this inner product. This implies that its action
on these modules is diagonalisable with real eigenvalues. A representation
theoretic argument shows that the reality of spectra of the Hamiltonian extends
to all other Temperley-Lieb representations. In particular, this result applies
to the celebrated $U_q(sl_2)$-invariant XXZ Hamiltonian, for all $q+q^{-1}\in
\mathbb R$.
