SUSY $SU(5)$ GUT chiral adjoint repersentations when you have a chiral superfield in the adjoint representation in your theory (i.e the Higgs $\Sigma_{24}$ in $SU(5)$), you can write it as a $5\times5$ Matrix: $(\Sigma_{24})_i^j=\Sigma^a (t^a)_i^j$ (sum over a), where the $t^a$ are the generatorns of the $SU(5)$. 
My question is: is it possible that the $\Sigma_a$ are real, or do they have to be complex for $SU(5)$ SUSY GUT?
I'm asking because some steps in the calculation of the minimum (vev) seem to need real $\Sigma_a$'s, since that would make $\Sigma_{24}$ Hermitian. But on the other side I'm not sure if real scalars would make problems with supersymmetry (degrees of freedom).
thanks

Update:
The problem with real $\Sigma_a$ is the following:
Suppose you want to write down the superfield $S$ in components. For one complex scalar $\Phi$ it would look as follows: $S=\Phi+\sqrt{2} \theta \Psi+\theta\theta F$. Looking at the degrees of freedome on-shell $\rightarrow dof(\Phi)=2, dof(\Psi)=2,dof(F)=0$
Now looking at the adjoint representation of $\Sigma$, wich is a 24-dimesional multiplett ($\Sigma_1,...,\Sigma_{24}$)(this are the $\Sigma_a$ from above, (btw. $\Sigma_a=\Sigma^a$, sorry for mixing notation)). Now each field in this supermultiplett has to be a superfield, like above $\Sigma_a=\Phi_a+\sqrt{2} \theta \Psi_a+\theta\theta F_a$. Counting now the degress of freedome reveals the problem: $\rightarrow dof(\Phi_a)=1, dof(\Psi_a)=2,dof(F_a)=0$. That shows that the fermionic dof. are not equal to the bosonic dof. wich contradicts SUSY.
Edit:


*

*$i,j$ in $\Sigma_i^j$ are matrix indices and run from 1 to 5.

*$a$ in $t^a$ and $\Sigma^a$ runs from 1 to 24 (24 = dimension of adjoint representation of $SU(5)$)
 A: In  any theory in  which supersymmetry  is  relevant down to  1 TeV,  the  supersymmetric  partners  of  the  leptons  are  spinless  bosons  with  the  right  $SU(2)\times U(1)$ properties  to  be  Higgs  mesons.  However,  for  various  reasons,  they  cannot  be.
The most  serious  objection  to  using  the  lepton  partners  as  Higgs  mesons  is  that  the partners  of  the  charge  $-1/3$  quarks  then  mediate  baryon  number  changing  interactions.  To  eliminate  this  difficulty, the  Higgs  doublets  must  be  part  of  a  different $SU(5)$  multiplet  whose  color  triplet  components  are  superheavy.  To  ensure  that 
baryon  number  is  conserved  except  by  the  usual  $SU(5)$  baryon  number  changing interactions,  we require  that the Higgs multiplets couple to pairs of quark  and lepton multiplets  just  as in  the usual $SU(5)$  model. 
This in turns saying that taking the representations of Higgs in the adjoint (for real) just like minimal $SU(5)$, is acceptable.
