  We measure the local anisotropy of numerically simulated strong Alfv\'enic
turbulence with respect to two local, physically relevant directions: along the
local mean magnetic field and along the local direction of one of the
fluctuating Elsasser fields. We find significant scaling anisotropy with
respect to both these directions: the fluctuations are "ribbon-like" ---
statistically, they are elongated along both the mean magnetic field and the
fluctuating field. The latter form of anisotropy is due to scale-dependent
alignment of the fluctuating fields. The intermittent scalings of the
$n$th-order conditional structure functions in the direction perpendicular to
both the local mean field and the fluctuations agree well with the theory of
Chandran et al. 2015, while the parallel scalings are consistent with those
implied by the critical-balance conjecture. We quantify the relationship
between the perpendicular scalings and those in the fluctuation and parallel
directions, and find that the scaling exponent of the perpendicular anisotropy
(i.e., of the aspect ratio of the Alfv\'enic structures in the plane
perpendicular to the mean magnetic field) depends on the amplitude of the
fluctuations. This is shown to be equivalent to the anticorrelation of
fluctuation amplitude and alignment at each scale. The dependence of the
anisotropy on amplitude is shown to be more significant for the anisotropy
between the perpendicular and fluctuation-direction scales than it is between
the perpendicular and parallel scales.
