We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to
positive characteristic. A key part of the proof is a new algebraically closed
field containing the field \kk(t_1,\dots,t_d) of rational functions over an
algebraically closed field \kk of prime characteristic. As a corollary, we
extend the tropical Bertini theorem of Maclagan and Yu to arbitrary
characteristic, which removes the characteristic dependence from the
d-connectivity result for tropical varieties from that paper
