AA Tau is the archetype for a class of stars with a peculiar periodic photometric variability thought to be related to a warped inner disk structure with a nearly edge-on viewing geometry. We present high resolution ($sim$0.2) ALMA observations of the 0.87 and 1.3~mm dust continuum emission from the disk around AA Tau. These data reveal an evenly spaced three-ringed emission structure, with distinct peaks at 0.34, 0.66, and 0.99, all viewed at a modest inclination of 59.1$^{circ}pm$0.3$^{circ}$ (decidedly not edge-on). In addition to this ringed substructure, we find non-axisymmetric features including a `bridge of emission that connects opposite sides of the innermost ring. We speculate on the nature of this `bridge in light of accompanying observations of HCO$^+$ and $^{13}$CO (J=3--2) line emission. The HCO$^+$ emission is bright interior to the innermost dust ring, with a projected velocity field that appears rotated with respect to the resolved disk geometry, indicating the presence of a warp or inward radial flow. We suggest that the continuum bridge and HCO$^+$ line kinematics could originate from gap-crossing accretion streams, which may be responsible for the long-duration dimming of optical light from AA Tau.
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