  We show that a class of compounds with $I$4/$mcm$ crystalline symmetry hosts
three-dimensional semi-Dirac fermions. Unlike the known two-dimensional
semi-Dirac points, the degeneracy of these three-dimensional semi-Dirac points
is not lifted by spin-orbit coupling due to the protection by a nonsymmorphic
symmetry -- screw rotation in the $a-b$ plane and a translation along the $c$
axis. This crystalline symmetry is found in tetragonal perovskite oxides,
realizable in thin films by epitaxial strain that results in
a$^0$a$^0$c$^-$-type octahedral rotation. Interestingly, with broken
time-reversal symmetry, two pairs of Weyl points emerge from the semi-Dirac
points within the Brillouin zone, and an additional lattice distortion leads to
enhanced intrinsic anomalous Hall effect. The ability to tune the Berry phase
by epitaxial strain can be useful in novel oxide-based electronic devices.
