We refine the recently introduced "Virtual VNA 3.0" technique to remove the need for coherent detection. The resulting "Virtual VNA 3.1" technique can unambiguously estimate the full scattering matrix of a non-reciprocal, linear, passive, time-invariant device under test (DUT) with $N$ monomodal ports using an $N_\mathrm{A}$-channel coherent wavefront generator and an $N_\mathrm{A}$-channel non-coherent detector, where $N_\mathrm{A}<N$. Waves are injected and received only via a fixed set of $N_\mathrm{A}$ "accessible" DUT ports while the remaining $N_\mathrm{S}$ "not-directly-accessible" DUT ports are terminated by a specific tunable load network. To resolve all ambiguities, an additional modified setup is required in which waves are injected and received via a known $2N_\mathrm{A}$-port system connected to the DUT's accessible ports. We experimentally validate our method for $N_\mathrm{A}=N_\mathrm{S}=4$ considering a non-reciprocal eight-port circuit as DUT. By eliminating the need for coherent detection, our work reduces the hardware complexity which may facilitate applications to large-scale or higher-frequency systems. Additionally, our work provides fundamental insights into the minimal requirements to fully and unambiguously characterize a non-reciprocal DUT.