  We explore non-Abelian strings in the r=N-1 vacuum of N=2 supersymmetric QCD
with the gauge group U(N) and N_f flavors of quarks (N_f\geq N), where r is the
number of condensed quarks. N=2 supersymmetry is broken down to N=1 by a small
mass term for the adjoint matter. We discover that the low-energy
two-dimensional theory on the string world-sheet receives nonperturbative
corrections from the bulk, through the bulk gaugino condensate. This is in
contradistinction with the r=N vacuum situation, in which nonperturbative
effects on the world sheet are determined by internal dynamics of the
world-sheet theory. The 2D-4D correspondence (the coincidence of spectra of
two-dimensional kinks and four-dimensional monopoles) remains valid in the BPS
sector. Nonperturbative bulk effects deforming the weighted CP model on the
world sheet are found by virtue of the method of resolvents suggested by
Gaiotto, Gukov and Seiberg for surface defects. In the r=N vacuum the gaugino
condensate in the bulk vanishes, and there are no "outside" nonperturbative
corrections on the world sheet.
