  We probe the magnetic field-induced Tomonaga-Luttinger liquid (TLL) state in
the bond-alternating spin-$1/2$ antiferromagnetic (AFM) chain compound
NaVOPO$_4$ using thermodynamic as well as local $\mu$SR and $^{31}$P NMR probes
down to milli-K temperatures in magnetic fields up to 14~T. The $\mu$SR and NMR
relaxation rates in the gapless TLL regime decay slowly following
characteristic power-law behaviour, enabling us to directly determine the
interaction parameter $K$ as a function of the magnetic field. These estimates
are cross-checked using magnetization and specific heat data. The
field-dependent $K$ lies in the range of $0.4 < K < 1$ and indicates repulsive
nature of interactions between the spinless fermions, in line with the
theoretical predictions. This renders NaVOPO$_4$ the first experimental
realization of TLL with repulsive fermionic interactions in hitherto studied
$S=1/2$ bond-alternating AFM-AFM chain compounds.
