Context. Unveiling the structure of the disks around intermediate-mass pre-main-sequence stars (Herbig Ae/Be stars) is essential for our understanding of the star and planet formation process. In particular, models predict that in the innermost AU around the star, the dust disk forms a "puffed-up" inner rim, which should result in a strongly asymmetric brightness distribution for disks seen under intermediate inclination.Aims. Our aim is to constrain the sub-AU geometry of the inner disk around the Herbig Ae star R CrA and search for the predicted asymmetries. Methods. Using the VLTI/AMBER long-baseline interferometer, we obtained 24 near-infrared (H-and K-band) spectrointerferometric observations on R CrA. Observing with three telescopes in a linear array configuration, each data set samples three equally spaced points in the visibility function, providing direct information about the radial intensity profile. In addition, the observations cover a wide position angle range (∼97 • ), also probing the position angle dependence of the source brightness distribution. Results. In the derived visibility function, we detect the signatures of an extended (Gaussian FWHM ∼ 25 mas) and a compact component (Gaussian FWHM ∼ 5.8 mas), with the compact component contributing about two-thirds of the total flux (both in H-and K-band). The brightness distribution is highly asymmetric, as indicated by the strong closure phases (up to ∼40 • ) and the detected position angle dependence of the visibilities and closure phases. To interpret these asymmetries, we employ various geometric as well as physical models, including a binary model, a skewed ring model, and a puffed-up inner rim model with a vertical or curved rim shape. For the binary and vertical rim model, no acceptable fits could be obtained. On the other hand, the skewed ring model and the curved puffed-up inner rim model allow us to simultaneously reproduce the measured visibilities and closure phases. From these models we derive the location of the dust sublimation radius (∼0.4 AU), the disk inclination angle (∼35 • ), and a north-south disk orientation (PA ∼ 180−190 • ). Our curved puffed-up rim model can reproduce reasonably well the interferometric observables and the SED and suggests a luminosity of ∼29 L and the presence of relatively large ( > ∼ 1.2 μm) Silicate dust grains. Our study also reveals discrepancies between the measured interferometric observables and the puffed-up inner rim models, providing important constraints for future refinements of these theoretical models. Perpendicular to the disk, two bow shock-like structures appear in the associated reflection nebula NGC 6729, suggesting that the detected sub-AU size disk is the driving engine of a large-scale outflow. Conclusions. Detecting, for the first time, strong non-localized asymmetries in the inner regions of a Herbig Ae disk, our study supports the existence of a puffed-up inner rim in YSO disks.