Motivated by the temporal dynamics identified in the $1+1$ Carroll-Schrödinger theory, we derive a post-Carrollian Schrödinger dynamics in flat Klein space with signature (2, 2). Starting from the tachyonic Klein-Gordon equation in double-polar coordinates and removing a spacelike carrier, the spatial radius behaves as an effective evolution parameter, whereas the temporal two-plane $(t_1,t_2)$ serves as the equal-radius configuration space. The additional time direction supplies an $SO(2)$ temporal angular momentum $J$, produces temporal vortex sectors, and gives the centrifugal contribution to the post-Carrollian momentum $P_{\mathrm{PC}}=E_{\tau }^2/(2M_{\mathrm{eff}})+J^2/(2M_{\mathrm{eff}}{\tau }^2)$ in the Hamilton-Jacobi limit. We determine the regular Bessel modes, Gaussian packets, oscillator spectrum, radial $SU(1,1)$ tower, equal-$r$ continuity equation, $\mathfrak{sch}(2)$ symmetry algebra, radial-ordered propagator, and the metaplectic organization of the quadratic sectors. Starting from a minimally coupled complex scalar-$U(1)$ parent action, we derive the Maxwell-matter equations and their Bianchi constraints, identify a regular finite-interaction branch with $q_{\epsilon }=q/\epsilon$, and obtain the corresponding gauge-covariant Carroll-Schrödinger equation together with its sourced Maxwell responses and optional temporal gauge backgrounds. As a curved extension, we derive a branch-dependent carrier reduction and apply it to an illustrative $SO(2,1)$-symmetric Kleinian Schwarzschild exterior.

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