diff --git a/Project.toml b/Project.toml
index 06bf43758..54ad35131 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,12 +1,13 @@
 name = "ChainRules"
 uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
-version = "1.11.6"
+version = "1.12.0"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
 Compat = "34da2185-b29b-5c13-b0c7-acf172513d20"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+RealDot = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
 
 [compat]
@@ -15,6 +16,7 @@ ChainRulesTestUtils = "1"
 Compat = "3.35"
 FiniteDifferences = "0.12.8"
 JuliaInterpreter = "0.8"
+RealDot = "0.1"
 StaticArrays = "1.2"
 julia = "1"
 
diff --git a/src/ChainRules.jl b/src/ChainRules.jl
index b951a0e85..f2806fd3d 100644
--- a/src/ChainRules.jl
+++ b/src/ChainRules.jl
@@ -6,6 +6,7 @@ using Compat
 using LinearAlgebra
 using LinearAlgebra.BLAS
 using Random
+using RealDot: realdot
 using Statistics
 
 # Basically everything this package does is overloading these, so we make an exception
diff --git a/src/rulesets/Base/base.jl b/src/rulesets/Base/base.jl
index 5b9237bcf..4118aa37e 100644
--- a/src/rulesets/Base/base.jl
+++ b/src/rulesets/Base/base.jl
@@ -78,7 +78,7 @@ end
 
 function frule((_, Δz), ::typeof(hypot), z::Complex)
     Ω = hypot(z)
-    ∂Ω = _realconjtimes(z, Δz) / ifelse(iszero(Ω), one(Ω), Ω)
+    ∂Ω = realdot(z, Δz) / ifelse(iszero(Ω), one(Ω), Ω)
     return Ω, ∂Ω
 end
 
diff --git a/src/rulesets/Base/fastmath_able.jl b/src/rulesets/Base/fastmath_able.jl
index d1839518a..2ef7a1f9c 100644
--- a/src/rulesets/Base/fastmath_able.jl
+++ b/src/rulesets/Base/fastmath_able.jl
@@ -68,7 +68,7 @@ let
             Ω = abs(x)
             # `ifelse` is applied only to denominator to ensure type-stability.
             signx = x isa Real ? sign(x) : x / ifelse(iszero(x), one(Ω), Ω)
-            return Ω, _realconjtimes(signx, Δx)
+            return Ω, realdot(signx, Δx)
         end
 
         function rrule(::typeof(abs), x::Union{Real, Complex})
@@ -82,7 +82,7 @@ let
 
         ## abs2
         function frule((_, Δz), ::typeof(abs2), z::Union{Real, Complex})
-            return abs2(z), 2 * _realconjtimes(z, Δz)
+            return abs2(z), 2 * realdot(z, Δz)
         end
 
         function rrule(::typeof(abs2), z::Union{Real, Complex})
@@ -146,7 +146,7 @@ let
         ) where {T<:Union{Real,Complex}}
             Ω = hypot(x, y)
             n = ifelse(iszero(Ω), one(Ω), Ω)
-            ∂Ω = (_realconjtimes(x, Δx) + _realconjtimes(y, Δy)) / n
+            ∂Ω = (realdot(x, Δx) + realdot(y, Δy)) / n
             return Ω, ∂Ω
         end
 
diff --git a/src/rulesets/Base/mapreduce.jl b/src/rulesets/Base/mapreduce.jl
index 41cbca20c..0a658454c 100644
--- a/src/rulesets/Base/mapreduce.jl
+++ b/src/rulesets/Base/mapreduce.jl
@@ -111,13 +111,13 @@ function frule(
     ẋ = unthunk(Δx)
     y = sum(abs2, x; dims=dims)
     ∂y = if dims isa Colon
-        2 * real(dot(x, ẋ))
+        2 * realdot(x, ẋ)
     elseif VERSION ≥ v"1.2" # multi-iterator mapreduce introduced in v1.2
         mapreduce(+, x, ẋ; dims=dims) do xi, dxi
-            2 * _realconjtimes(xi, dxi)
+            2 * realdot(xi, dxi)
         end
     else
-        2 * sum(_realconjtimes.(x, ẋ); dims=dims)
+        2 * sum(realdot.(x, ẋ); dims=dims)
     end
     return y, ∂y
 end
diff --git a/src/rulesets/Base/utils.jl b/src/rulesets/Base/utils.jl
index 30bbdd47e..9dd991742 100644
--- a/src/rulesets/Base/utils.jl
+++ b/src/rulesets/Base/utils.jl
@@ -1,10 +1,3 @@
-# real(conj(x) * y) avoiding computing the imaginary part if possible
-@inline _realconjtimes(x, y) = real(conj(x) * y)
-@inline _realconjtimes(x::Complex, y::Complex) = muladd(real(x), real(y), imag(x) * imag(y))
-@inline _realconjtimes(x::Real, y::Complex) = x * real(y)
-@inline _realconjtimes(x::Complex, y::Real) = real(x) * y
-@inline _realconjtimes(x::Real, y::Real) = x * y
-
 # imag(conj(x) * y) avoiding computing the real part if possible
 @inline _imagconjtimes(x, y) = imag(conj(x) * y)
 @inline function _imagconjtimes(x::Complex, y::Complex)
diff --git a/src/rulesets/LinearAlgebra/blas.jl b/src/rulesets/LinearAlgebra/blas.jl
index 509e15847..c0f90a31d 100644
--- a/src/rulesets/LinearAlgebra/blas.jl
+++ b/src/rulesets/LinearAlgebra/blas.jl
@@ -41,7 +41,7 @@ function frule((_, Δx), ::typeof(BLAS.nrm2), x)
     ∂Ω = if x isa Real
         BLAS.dot(x, Δx) / s
     else
-        sum(y -> _realconjtimes(y...), zip(x, Δx)) / s
+        sum(y -> realdot(y...), zip(x, Δx)) / s
     end
     return Ω, ∂Ω
 end
@@ -72,7 +72,7 @@ end
 
 function frule((_, Δx), ::typeof(BLAS.asum), x)
     ∂Ω = sum(zip(x, Δx)) do (xi, Δxi)
-        return _realconjtimes(_signcomp(xi), Δxi)
+        return realdot(_signcomp(xi), Δxi)
     end
     return BLAS.asum(x), ∂Ω
 end
diff --git a/src/rulesets/LinearAlgebra/factorization.jl b/src/rulesets/LinearAlgebra/factorization.jl
index f5342bfad..d7530a16e 100644
--- a/src/rulesets/LinearAlgebra/factorization.jl
+++ b/src/rulesets/LinearAlgebra/factorization.jl
@@ -347,7 +347,7 @@ function _eigen_norm_phase_fwd!(∂V, A, V)
     @inbounds for i in axes(V, 2)
         v, ∂v = @views V[:, i], ∂V[:, i]
         # account for unit normalization
-        ∂c_norm = -real(dot(v, ∂v))
+        ∂c_norm = -realdot(v, ∂v)
         if eltype(V) <: Real
             ∂c = ∂c_norm
         else
diff --git a/src/rulesets/LinearAlgebra/norm.jl b/src/rulesets/LinearAlgebra/norm.jl
index c2114c867..2cc69355a 100644
--- a/src/rulesets/LinearAlgebra/norm.jl
+++ b/src/rulesets/LinearAlgebra/norm.jl
@@ -14,7 +14,7 @@ function frule((_, ẋ), ::typeof(norm), x::Number, p::Real)
         zero(real(x)) * zero(real(Δx))
     else
         signx = x isa Real ? sign(x) : x * pinv(y)
-        _realconjtimes(signx, Δx)
+        realdot(signx, Δx)
     end
     return y, ∂y
 end
@@ -235,7 +235,7 @@ function rrule(::typeof(LinearAlgebra.norm2), x::AbstractArray{<:Number})
 end
 
 function _norm2_forward(x, Δx, y)
-    ∂y = real(dot(x, Δx)) * pinv(y)
+    ∂y = realdot(x, Δx) * pinv(y)
     return ∂y
 end
 function _norm2_back(x, y, Δy)
@@ -280,7 +280,7 @@ function rrule(::typeof(normalize), x::AbstractVector{<:Number})
     LinearAlgebra.__normalize!(y, nrm)
     function normalize_pullback(ȳ)
         Δy = unthunk(ȳ)
-        ∂x = (Δy .- real(dot(y, Δy)) .* y) .* pinv(nrm)
+        ∂x = (Δy .- realdot(y, Δy) .* y) .* pinv(nrm)
         return (NoTangent(), ∂x)
     end
     normalize_pullback(::ZeroTangent) = (NoTangent(), ZeroTangent())
diff --git a/src/rulesets/LinearAlgebra/symmetric.jl b/src/rulesets/LinearAlgebra/symmetric.jl
index 573d4fcf4..67693575e 100644
--- a/src/rulesets/LinearAlgebra/symmetric.jl
+++ b/src/rulesets/LinearAlgebra/symmetric.jl
@@ -218,7 +218,7 @@ function frule(
     # diag(U' * tmp) without computing matrix product
     ∂λ = similar(λ)
     @inbounds for i in eachindex(λ)
-        ∂λ[i] = @views real(dot(U[:, i], tmp[:, i]))
+        ∂λ[i] = @views realdot(U[:, i], tmp[:, i])
     end
     return λ, ∂λ
 end