  We consider scalar field theory defined over a direct product of the real and
$p$-adic numbers. An adjustable dynamical scaling exponent $z$ enters into the
microscopic lagrangian, so that the Gaussian theories provide a line of fixed
points. We argue that at $z=1/3$, a branch of Wilson-Fisher fixed points joins
onto the line of Gaussian theories. We compute standard critical exponents at
the Wilson-Fisher fixed points in the region where they are perturbatively
accessible, including a loop correction to the dynamical critical exponent. We
show that the classical propagator contains oscillatory behavior in the real
direction, though the amplitude of these oscillations can be made exponentially
small without fine-tuning parameters of the theory. Similar oscillatory
behavior emerges in Fourier space from two-loop corrections, though again it
can be highly suppressed. We also briefly consider compact $p$-adic extra
dimensions, showing in non-linear, classical, scalar field theories that a form
of consistent truncation allows us to retain only finitely many Kaluza-Klein
modes in an effective theory formulated on the non-compact directions.
