Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:

1. it's not isomorphic to its dual (in which case we call it 'complex')

2. it has a nondegenerate symmetric bilinear form (in which case we call it 'real')

3. it has a nondegenerate antisymmetric bilinear form (in which case we call it 'quaternionic')

It's 'real' in this sense iff it's the complexification of a representation on a real vector space, and it's 'quaternionic' in this sense iff it's the underlying complex representation of a representation on a quaternionic vector space.

Offhand, I know just four compact Lie groups where all continuous irreducible representations are either real or quaternionic:

1. the group Z/2

2. the trivial group

3. the group SU(2)

4. the group SO(3)

Note that I'm so desperate for examples that I'm including 0-dimensional compact Lie groups, i.e. finite groups!

Note that 1) is the group of unit-norm real numbers, 2) is a group covered by that, 3) is the group of unit-norm quaternions, and 4) is a group covered by that. That probably explains why these are all the examples I know. For 2) and 4), all the continuous irreducible representations are in fact real.

What are all the examples?