  Garside theory emerged from the study of Artin groups and their
generalizations. Finite-type Artin groups admit two types of interval Garside
structures corresponding to their standard and dual presentations. Concerning
affine Artin groups, Digne established interval Garside structures for two
families of these groups by using their dual presentations. Recently, McCammond
established that none of the remaining dual presentations (except for one
additional case) correspond to interval Garside structures. In this paper,
shifting attention from dual presentations to other nice presentations for the
affine Artin group of type $\tilde{A}$ discovered by Shi and Corran--Lee--Lee,
I will construct interval Garside structures related to this group. This
construction is the first successful attempt to establish interval Garside
structures not related to the dual presentations in the case of affine Artin
groups.

Group Theory