We present the first joint inference of standard sirens and gravitational wave weak lensing by filtering of the same dataset. We imagine a post-LISA scenario emerging around the late 2030s when LISA will have accumulated a number of detections at high redshift, LIGO-VIRGO will have finished observing at low redshift, and Einstein Telescope will have started making new observations out to redshifts possibly overlapping with LISA. Euclid and other cosmological probes will have provided constraints at the percent level by then, but will have mostly exhausted their ability to improve any further. We derive forecasts assuming ∼1 deg-2 detected sources, in conjunction with a spectroscopic follow-up (e.g., by Euclid, DESI, or ATHENA). Thanks to the statistical power of standard sirens as a geometry probe—lifting key degeneracies in the gravitational wave weak lensing—and no external priors assumed, the constraints on dark matter and its clustering, namely Ωm and σ8, could be achieved to 2% and 3%. The Hubble constant could be constrained to better than 1% in all cases; the dark energy density, ΩΛ, to 2%; the curvature, ΩK, to 0.02; and the amplitude and spectral tilt of the scalar fluctuations, ln (1010As) and ns, to 2% and 7%. As a completely independent cosmological probe, with fewer calibration requirements, the joint inference of standard sirens and gravitational wave weak lensing might help solve the tensions currently observed between other cosmological probes, such as the CMB, galaxy lensing, and type-Ia supernovae, and distinguish between residual systematics and new physics.