Laser-guide-star multiconjugate adaptive optics (MCAO) systems require natural guide stars (NGS) to measure tilt and tilt-anisoplanatism modes. Making optimal use of the limited number of photons coming from such, generally dim, sources is mandatory to obtain reasonable sky coverage, i.e., the probability of finding asterisms amenable to NGS wavefront (WF) sensing for a predefined WF error budget. This paper presents a Strehl-optimal (minimum residual variance) spatiotemporal reconstructor merging principles of modal atmospheric tomography and optimal stochastic control theory. Simulations of NFIRAOS, the first light MCAO system for the thirty-meter telescope, using $\sim 500$ typical NGS asterisms, show that the minimum-variance (MV) controller delivers outstanding results, in particular for cases with relatively dim stars (down to magnitude 22 in the H-band), for which low-temporal frame rates (as low as 16 Hz) are required to integrate enough flux. Over all the cases tested $\sim 21\mathrm{nm}$ rms median improvement in WF error can be achieved with the MV compared to the current baseline, a type-II controller based on a double integrator. This means that for a given level of tolerable residual WF error, the sky coverage is increased by roughly 10%, a quite significant figure. The improvement goes up to more than 20% when compared with a traditional single-integrator controller.