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, Pr2Ir2O7 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 the [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 Pr2Ir2O7. We propose a strain-induced enhancement of the onset temperature for AHE in thin films of Pr2Ir2O7 and additional experiments for studying competing orders in the vicinity of the metal-insulator transition. In addition to providing a theory for competing orders and magnetic properties of Pr2Ir2O7, the theoretical framework developed in this work should also be useful for a better understanding of competing multipolar orders in other correlated materials.