  We investigate how SL(2,Z) duality is realized in nonrelativistic type IIB
superstring theory, which is a self-contained corner of relativistic string
theory. Within this corner, we realize manifestly SL(2,Z)-invariant
(p,q)-string actions. The construction of these actions imposes a branching
between strings of opposite charges associated with the two-form fields. The
branch point is determined by these charges and the axion background field.
Both branches must be incorporated in order to realize the full SL(2,Z) group.
Besides these string actions, we also construct D-instanton and D3-brane
actions that manifestly realize the branched SL(2,Z) symmetry.
