Let X be  a  hausdorff space such that every real  vector  bundle on X is  summand of  a trivial bundle. Does  it implies that X is  homotopy  equivalent to a  compact  Hausdorf space? This  question is  a "compact  version" of the  following question;

http://mathoverflow.net/questions/149138/paracompactness-and-inner-product-on-vector-bundles