Fermionic linear optics (FLO) with Gaussian resources is known to be efficiently classically simulable. We show that this changes for fermions with internal degrees of freedom once the circuit is} equipped with mid-circuit number monitoring and classical feedforward. In our architecture, the measurement record routes the selected blocks into a fixed-order Bell-fusion pairing geometry. On the level of the classical description, the permutation sum then no longer collapses to a single determinant or Pfaffian. Each accepted branch expands as a signed sum of path-ordered products of typically non-commuting dressed blocks, and branch amplitudes are matrix elements of the resulting non-commutative trace polynomials. Numerically, we observe Porter-Thomas statistics for the conditional branch distribution and rapid growth of the minimal order-respecting matrix product operator bond dimension. Under reasonable complexity assumptions, these results identify mid-circuit measurement-induced non-commutativity as a route to sampling hardness for noninteracting fermions, with the non-Gaussianity confined to a fixed preparation-and-readout interface rather than to the coherent FLO evolution.