RFC: Rules for real-to-complex and complex-to-real functions

Consider a case where we have a function f: ℝᵐ → ℂʳ → ℂˢ → ℝⁿ = ℝᵐ → ℝⁿ, which we can write as f = f₃ ∘ f₂ ∘ f₁.
Typically f₁ will produce a complex output by adding, subtracting, multiplying or dividing the real by a complex number
or by calling promote, complex, Complex or cis.
Typically f₃ will produce a real output by calling a non-holomorphic function like real, imag, abs, abs2, hypot, or angle.

From https://github.com/JuliaDiff/ChainRulesCore.jl/pull/167, the fact that there are complex intermediates to f is just an implementation detail. We could have defined f: ℝᵐ → ℝ²ʳ → ℝ²ˢ → ℝⁿ, and the pushforwards and pullbacks of this new f should behave the same.

Since in general tangents are derivatives of a primal wrt a real, and co-tangents are derivatives of a real wrt a primal,
the pushforward through f₁: ℝᵐ → ℂʳ should produce a complex tangent, while the pushforward through f₃: ℂˢ → ℝⁿ should produce a real tangent.
Conversely, the pullback through f₃ should produce a complex cotangent, and the pullback through f₁ should produce a real cotangent.

The pushforward case is pretty easy to handle. We can 1) assume that a non-sensical tangent will not be passed and do nothing special (i.e. assume upstream AD did the right thing) or 2) define custom frules that ensure that the produced tangent of unary functions f₃(::Complex)::Real is real.

The pullback case is more complicated. Right now e.g. in Zygote, unless you create a complex number from reals by calling complex, you'll end up pulling back complex numbers through the initial real part of your program, which not only is wasteful but could break assumptions of the rrules of upstream functions. I propose for the binary functions f₁ adding custom rrules for f₁(::Real, ::Complex)::Complex and f₁(::Complex, ::Real)::Complex to ensure that the co-tangent pulled back to a real primal is actually real.

This came up a point of discussion in JuliaDiff/ChainRules.jl#196, and I would appreciate feedback so we can clarify our conventions here.

Start by reading the discussion in JuliaDiff/ChainRulesCore.jl#167 and JuliaDiff/ChainRules.jl#196, then compare the proposed behavior for real/complex pushforwards and pullbacks with the existing conventions. Done means the project has an agreed rule set and an actionable implementation scope for the relevant binary and unary functions.