The Madden-Julian Oscillation (MJO) is the dominant component of tropical intraseasonal variability, and a theory explaining its structure and successful numerical simulation remains a major challenge. A successful model for the MJO should have a propagation speed of 4-7 m/s predicted by theory; a wavenumber-2 or -3 structure for the planetary-scale, low-frequency envelope with distinct active and inactive phases of deep convection; an intermittent turbulent chaotic multiscale structure within the planetary envelope involving embedded westward- and eastward-propagating deep convection events; and qualitative features of the low-frequency envelope from the observational record regarding, e.g., its zonal flow structure and heating. Here, such an MJO analog is produced by using the recent multicloud model of Khouider and Majda in an appropriate intraseasonal parameter regime for flows above the equator so that rotation is ignored. Key features of the multicloud model are (i) systematic low-level moisture convergence with retained conservation of vertically integrated moist static energy, and (ii) the use of three cumulus cloud types (congestus, stratiform, and deep convective) together with their differing vertical heating structures. Besides all of the above structure in the MJO analog waves, there are accurate predictions of the phase speed from linear theory and transitions from weak, regular MJO analog waves to strong, multiscale MJO analog waves as climatological parameters vary. With all of this structure in a simplified context, these models should be useful for MJO predictability studies in a fashion akin to the Lorenz 96 model for the midlatitude atmosphere.