I tried solving over the Reals with MemoryConstrained[] constraint of 1GB. It aborted. The solutions over the complexes are 17MB in size and do not contain complex numbers. The real ones, if you can get the computation to finish, are likely to be much more complicated, and I wonder how useful they will be.

solutions = Solve[equations, vars];
solutions // ByteCount
(*  17399816  *)

Cases[solutions, _Complex, Infinity] // DeleteDuplicates
(*  {}  *)

Below is a way to get conditions on the solutions so that each solution is real. We can see there are usually four conditions on each, except the first two, which are explicit integer solutions. Maybe you can make use of that. (I realize these may not produce all possibilities, since it is possible for two complex terms to add up to a real value.)

DeleteDuplicates@Cases[#, Power[b_, _Rational] :> b >= 0, Infinity] & /@ 
  solutions

Length@DeleteDuplicates@Cases[#, Power[b_, _Rational] :> b >= 0, Infinity] & /@ 
  solutions
(*  {0, 0, 4, 4, 4, 4, 4, 4, 4, 4}  *)

Two real trivial solutions are guaranteed:

equations /. {{X -> 0, Y -> 0, Z -> -1, M -> 0, A -> 0, 
   B -> 0}, {X -> 0, Y -> 0, Z -> 1, M -> 0, A -> 0, B -> 0}}

In general the situation is very complicated:

Solve[equations /. {\[Omega]a -> 2, \[Omega]c -> 1, \[Kappa]1 -> 
    2, \[Lambda]x -> 3, \[Lambda]y -> 1}, vars, Reals]

Hello Professor, I was wondering why we can immediately obtain an analytical solution when the real-number restriction is removed, even though all my variables are indeed not complex. Does adding the real-number constraint definitely mean that no analytical solution exists