  Starting with a N=4 supersymmetric Yang-Mills theory in four dimensions with
gauge group SU(3N) we perform an orbifold projection leading to a N=1
supersymmetric SU(N)^3 Yang-Mills theory with matter supermultiplets in
bifundamental representations of the gauge group, which is chiral and anomaly
free. Subsequently, we search for vacua of the projected theory which can be
interpreted as spontaneously generated twisted fuzzy spheres. We show that by
adding the appropriate soft supersymmetry breaking terms we can indeed reveal
such vacua. Three cases are studied, where the gauge group is spontaneously
broken further to the low-energy gauge groups SU(4)xSU(2)xSU(2), SU(4)^3 and
SU(3)^3. Such models behave in intermediate scales as higher-dimensional
theories with a finite Kaluza-Klein tower, while their low-energy physics is
governed by the corresponding zero-modes and exhibit chirality in the fermionic
sector. The most interesting case from the phenomenological point of view turns
out to be the SU(3)^3 unified theory, which has several interesting features
such as (i) it can be promoted to a finite theory, (ii) it breaks further
spontaneously first to the MSSM and then to SU(3)xU(1)_{em} due to its own
scalar sector, i.e. without the need of additional superfields and (iii) the
corresponding vacua lead to spontaneously generated fuzzy spheres.
