Consider a real Hilbert bundle E with structure group contained in a real UHF algebra \({\mathcal A}\subset L(H)\). Then E may be orientable or not depending on the ''type'' of \({\mathcal A}\). More precisely, we proved the following result on the homotopical structure of the group G(\({\mathcal A})\) of invertible elements: G(\({\mathcal A})\) is connected iff \(K_ 0({\mathcal A})\) contains \({\mathbb{Z}}()\), the group of dyadic rationals. If this holds then G(\({\mathcal A})\) is even simply connected.