We present the first missing mass spectrum of the unbound nucleus He10, measured at RIKEN using the Li11(d,He3) reaction at 50A MeV. He10 is believed to be a three-body He8+n+n resonance beyond the limit of nuclear binding. Our observation of a new decay branch, 6He+4n, and of a puzzling reduction of the Li11(d,He3)He10 cross section challenges this view. Moreover, our experiment shows a new trend in the evolution of the proton spectroscopic strength in Li isotopes deduced from the comparison of the (d,He3) cross sections on Li7,8,9,11 with theoretical predictions. We discuss new questions about physics beyond the limits of nuclear existence raised by these findings.