The classical Bertini theorem on irreducibility when
intersecting by hyperplanes is a standard part of the algebraic
geometry toolkit.  This was generalised recently, in characteristic
zero, by Fuchs, Mantova, and Zannier to a toric Bertini theorem for
subvarieties of an algebraic torus, with hyperplanes replaced by
subtori.  I will discuss joint work with Gandini, Hering, Mohammadi,
Rajchgot, Wheeler, and Yu in which we give a different proof of this
theorem that removes the characteristic assumption.  The proof
surprisingly hinges on better understanding algebraically closed
fields containing the field of rational functions in n variables,
which involve polyhedral constructions.  An application is a tropical
Bertini theorem.