  Strong electronic interactions and spin orbit coupling can be conducive for
realizing novel broken symmetry phases supporting quasiparticles with
nontrivial band topology. 227 pyrochlore iridates provide a suitable material
platform for studying such emergent phenomena where both topology and competing
orders play important roles. In contrast to the most members of this material
class, which are thought to display "all-in all-out" (AIAO) type magnetically
ordered low-temperature insulating ground states, Pr$_2$Ir$_2$O$_7$ remains
metallic while exhibiting "spin ice" (SI) correlations at low temperatures.
Additionally, this is the only 227 iridate compound, which exhibits a large
anomalous Hall effect (AHE) along [1,1,1] direction below 1.5 K, without
possessing any measurable magnetic moment. By focusing on the normal state of
227 iridates, described by a parabolic semimetal with quadratic band touching,
we use renormalization group analysis, mean-field theory, and phenomenological
Landau theory as three complementary methods to construct a global phase
diagram in the presence of generic local interactions among itinerant electrons
of Ir ions. While the global phase diagram supports several competing
multipolar orders, motivated by the phenomenology of 227 iridates we
particularly emphasize the competition between AIAO and SI orders and how it
can cause a mixed phase with "three-in one-out" (3I1O) spin configurations. In
terms of topological properties of Weyl quasiparticles of the 3I1O state, we
provide an explanation for the magnitude and the direction of the observed AHE
in Pr$_2$Ir$_2$O$_7$. We propose a strain induced enhancement of the onset
temperature for AHE in thin films of Pr$_2$Ir$_2$O$_7$ and additional
experiments for studying competing orders in the vicinity of the
metal-insulator transition.
