  We put forward a novel method for producing ultrarelativistic high-density
high-polarization positrons through a single-shot interaction of a strong laser
with a tilted solid foil. In our method, the driving laser ionizes the target,
and the emitted electrons are accelerated and subsequently generate abundant
$\gamma$ photons via the nonlinear Compton scattering, dominated by the laser.
These $\gamma$ photons then generate polarized positrons via the nonlinear
Breit-Wheeler process, dominated by a strong self-generated quasi-static
magnetic field $\mathbf{B}^{\rm S}$. We find that placing the foil at an
appropriate angle can result in a directional orientation of $\mathbf{B}^{\rm
S}$, thereby polarizing positrons. Manipulating the laser polarization
direction can control the angle between the $\gamma$ photon polarization and
$\mathbf{B}^{\rm S}$, significantly enhancing the positron polarization degree.
Our spin-resolved quantum electrodynamics particle-in-cell simulations
demonstrate that employing a laser with a peak intensity of about $10^{23}$
W/cm$^2$ can obtain dense ($\gtrsim$ 10$^{18}$ cm$^{-3}$) polarized positrons
with an average polarization degree of about 70\% and a yield of above 0.1 nC
per shot. Moreover, our method is feasible using currently available or
upcoming laser facilities and robust with respect to the laser and target
parameters. Such high-density high-polarization positrons hold great
significance in laboratory astrophysics, high-energy physics and new physics
beyond the Standard Model.
