diff --git a/Project.toml b/Project.toml
index a56827cad..30a618092 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRulesCore"
 uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
-version = "1.26.6"
+version = "1.26.7"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/src/projection.jl b/src/projection.jl
index 7ddadf11a..ddc797bd5 100644
--- a/src/projection.jl
+++ b/src/projection.jl
@@ -483,10 +483,20 @@ for (SymHerm, chk, fun) in
             sub_one = ProjectTo{project_type(sub)}(;
                 element=sub.element, axes=(sub.axes[1],)
             )
-            return Diagonal(sub_one(dx.diag))
+            return Diagonal(_project_diag(project, sub_one, dx.diag))
         end
     end
 end
+_project_diag(::ProjectTo{Symmetric}, sub_one, diag) = sub_one(diag)
+function _project_diag(::ProjectTo{Hermitian}, sub_one, diag)
+    dy = sub_one(diag)
+    # The diagonal of a Hermitian matrix is real
+    if eltype(diag) <: Real || eltype(dy) <: Real
+        return dy
+    else
+        return map(y -> oftype(y, real(y)), dy)
+    end
+end
 
 # Triangular
 for UL in (:UpperTriangular, :LowerTriangular, :UnitUpperTriangular, :UnitLowerTriangular) # UpperHessenberg
diff --git a/test/projection.jl b/test/projection.jl
index d77788b71..dee6ef146 100644
--- a/test/projection.jl
+++ b/test/projection.jl
@@ -314,6 +314,23 @@ struct NoSuperType end
         # an experiment with allowing subspaces which aren't subtypes
         @test psymm(Diagonal([1, 2, 3])) isa Diagonal{Float64}
         @test pupp(Diagonal([1, 2, 3 + 4im])) isa Diagonal{Float64}
+
+        @testset "Diagonal input to Hermitian has real diagonal" begin
+            dx = Diagonal([1 + 2im, 3im, 4.0])
+            @test @inferred(pherm(dx)) == Diagonal([1.0, 0.0, 4.0])
+            @test pherm(dx) isa Diagonal{ComplexF64}
+            @test pherm(dx) == pherm(Matrix(dx))
+            @test @inferred(pherm(Diagonal([1.0, 2.0, 3.0]))) == Diagonal([1.0, 2.0, 3.0])
+            @test pherm(Diagonal([1.0, 2.0, 3.0])) isa Diagonal{ComplexF64}
+            pherm32 = ProjectTo(Hermitian(rand(ComplexF32, 3, 3)))
+            @test @inferred(pherm32(dx)) isa Diagonal{ComplexF32}
+            @test @inferred(ProjectTo(Hermitian(rand(3, 3)))(dx)) ==
+                Diagonal([1.0, 0.0, 4.0])
+            @test_throws DimensionMismatch pherm(Diagonal(rand(ComplexF64, 2)))
+
+            # Symmetric keeps a complex diagonal
+            @test @inferred(ProjectTo(Symmetric(rand(ComplexF64, 3, 3)))(dx)) == dx
+        end
     end
 
     @testset "LinearAlgebra: sparse structured matrices" begin
@@ -499,6 +516,14 @@ struct NoSuperType end
 
         psymm = ProjectTo(Symmetric(rand(10^3, 10^3)))
         @test 0 == @ballocated $psymm(dx) setup = (dx = Symmetric(rand(10^3, 10^3)))
+
+        pherm = ProjectTo(Hermitian(rand(10^3, 10^3)))
+        @test 0 == @ballocated $pherm(dx) setup = (dx = Diagonal(rand(10^3)))
+
+        # a real tangent onto complex Hermitian is copied only once
+        pherm_c = ProjectTo(Hermitian(rand(ComplexF64, 10^3, 10^3)))
+        @test 2 * sizeof(ComplexF64) * 10^3 >
+            @ballocated $pherm_c(dx) setup = (dx = Diagonal(rand(10^3)))
     end
 
     @testset "#407" begin