  Quantum coherence and quantum entanglement represent two fundamental features
of non-classical systems that can each be characterized within an operational
resource theory. In this paper, we unify the resource theories of entanglement
and coherence by studying their combined behavior in the operational setting of
local incoherent operations and classical communication (LIOCC). Specifically
we analyze the coherence and entanglement trade-offs in the tasks of state
formation and resource distillation. For pure states we identify the minimum
coherence-entanglement resources needed to generate a given state, and we
introduce a new LIOCC monotone that completely characterizes a state's optimal
rate of bipartite coherence distillation. This result allows us to precisely
quantify the difference in operational powers between global incoherent
operations, LIOCC, and local incoherent operations \textit{without} classical
communication. Finally, a bipartite mixed state is shown to have distillable
entanglement if and only if entanglement can be distilled by LIOCC, and we
strengthen the well-known Horodecki criterion for distillability.
