Tuning between a fractional topological insulator and competing phases at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math>

We study a spinful, time-reversal symmetric lowest Landau level model for a flatband quantum spin Hall system at total filling fraction <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>ν</a:mi> <a:mi mathvariant="normal">T</a:mi> </a:msub> <a:mo>=</a:mo> <a:mn>2</a:mn> <a:mo>/</a:mo> <a:mn>3</a:mn> </a:mrow> </a:math> . Such models are relevant, e.g., for spin-valley locked moiré transition metal dichalcogenides. The opposite Chern number of the two spins hinders the formation of a quantum Hall ferromagnet, instead favoring other phases. We study the phase diagram in dependence on different short-range Haldane pseudopotentials <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi>v</c:mi> <c:mi>m</c:mi> </c:msub> </c:math> and uncover several phases: a fractional topological insulator, a phase-separated state, a spin-polarized fractional quantum Hall state, and the partially particle-hole transformed Halperin (111) state. The effect of the pseudopotentials <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>v</d:mi> <d:mi>m</d:mi> </d:msub> </d:math> depends on the parity of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>m</e:mi> </e:math> , the relative angular momentum.