We present a model that includes the production of arbitrarily many jets in lepton-hadron events, using the leading log formalism for parton shower evolution. The main problem encountered here, which has not previously been illuminated by studies ofe+e− annihilation or Drell-Yan/Z0/W± production, is the choice of kinematics in the space-like shower evolution. In our preferred solution, the standard definition of Bjorkenx is preserved during the construction of initial and final state showers—a nontrivial constraint. The resulting model is described in detail, including some first investigations of its properties.