  We reconstruct the two-dimensional (2D) matter distributions in 20 high-mass
galaxy clusters selected from the CLASH survey by using the new approach of
performing a joint weak lensing analysis of 2D shear and azimuthally averaged
magnification measurements. This combination allows for a complete analysis of
the field, effectively breaking the mass-sheet degeneracy. In a Bayesian
framework, we simultaneously constrain the mass profile and morphology of each
individual cluster assuming an elliptical Navarro-Frenk-White halo
characterized by the mass, concentration, projected axis ratio, and position
angle of the projected major axis.. We find that spherical mass estimates of
the clusters from azimuthally averaged weak-lensing measurements in previous
work are in excellent agreement with our results from a full 2D analysis.
Combining all 20 clusters in our sample, we detect the elliptical shape of
weak-lensing halos at the $5\sigma$ significance level within a scale of
2Mpc$/h$. The median projected axis ratio is $0.67\pm 0.07$ at a virial mass of
$M_\mathrm{vir}=(15.2\pm 2.8) \times 10^{14} M_\odot$, which is in agreement
with theoretical predictions of the standard collisionless cold dark matter
model. We also study misalignment statistics of the brightest cluster galaxy,
X-ray, thermal Sunyaev-Zel'dovich effect, and strong-lensing morphologies with
respect to the weak-lensing signal. Among the three baryonic tracers studied
here, we find that the X-ray morphology is best aligned with the weak-lensing
mass distribution, with a median misalignment angle of $21\pm 7$ degrees. We
also conduct a stacked quadrupole shear analysis assuming that the X-ray major
axis is aligned with that of the projected mass distribution. This yields a
consistent axis ratio of $0.67\pm 0.10$, suggesting again a tight alignment
between the intracluster gas and dark matter.
