There are three kinds of representations: real, complex, and pseudoreal. A complex representation is not equivalent to its conjugate, and a real one is, which is pretty straightforward. A pseudoreal representation is also equivalent to its conjugate, but the change of basis matrix that relates them has some funny properties. (Note that these definitions are independent of the terms 'real' and 'complex' in math; all representations in quantum mechanics are 'complex' in the math sense.)
There's a clear physical meaning of a complex representation, i.e. that particles that transform in these representations are not the same as their antiparticles. But I can't find any simple physical meaning for pseudoreality that distinguishes it from reality; it looks to me to be a fairly arbitrary distinction and I don't even know why we would want to make that distinction on mathematical grounds. How should I think about reality and pseudoreality physically?