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The Kitaev honeycomb model has been suggested as a useful starting point todescribe the honeycomb iridates. However, the zigzag ordering seen in Na2IrO3and the magnetic transition in Li2IrO3 have raised questions to theirconnection to the Kitaev model and to the microscopic origin of these magneticstates, given their structural similarities. Here we study how the magneticphases near the Kitaev spin liquid are affected by the inclusion of trigonaldistortion of the oxygen octahedra within a purely nearest neighbour model.Using a combination classical analysis and exact diagonalization we show thatnear the ferromagnetic Kitaev limit a small amount of trigonal distortion, asfound in Na2IrO3, stabilizes a zigzag phase. Decreasing the distortiondestabilizes the zigzag phase toward a spiral phase that may be relevant forLi2IrO3. Using semi-classical spin-wave calculations we show that this regimeis qualitatively consistent with experimentally known features of the dynamicalstructure factor in Na2IrO3 and speculate on implications for Li2IrO3.

Trigonal distortion in the honeycomb iridates: Proximity of zigzag and spiral phases in Na2IrO3