  We present a novel classical algorithm designed to learn the stabilizer group
-- namely the group of Pauli strings for which a state is a $\pm 1$ eigenvector
-- of a given Matrix Product State (MPS). The algorithm is based on a clever
and theoretically grounded biased sampling in the Pauli (or Bell) basis. Its
output is a set of independent stabilizer generators whose total number is
directly associated with the stabilizer nullity, notably a well-established
nonstabilizer monotone. We benchmark our method on $T$-doped states randomly
scrambled via Clifford unitary dynamics, demonstrating very accurate estimates
up to highly-entangled MPS with bond dimension $\chi\sim 10^3$. Our method,
thanks to a very favourable scaling $\mathcal{O}(\chi^3)$, represents the first
effective approach to obtain a genuine magic monotone for MPS, enabling
systematic investigations of quantum many-body physics out-of-equilibrium.
