diff --git a/Project.toml b/Project.toml
index ff52dd8bb..ddb2fe3fd 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRules"
 uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
-version = "1.73.0"
+version = "1.73.1"
 
 [deps]
 Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
diff --git a/src/rulesets/LinearAlgebra/symmetric.jl b/src/rulesets/LinearAlgebra/symmetric.jl
index cdadca6c3..2e713c2ed 100644
--- a/src/rulesets/LinearAlgebra/symmetric.jl
+++ b/src/rulesets/LinearAlgebra/symmetric.jl
@@ -381,7 +381,19 @@ function _matfun(f, A::LinearAlgebra.RealHermSymComplexHerm)
     df_dλ = last.(unthunk.(fλ_df_dλ))
     fA = (U * Diagonal(fλ)) * U'
     Y = if eltype(A) <: Real
-        Symmetric(fA)
+        if eltype(fλ) <: Complex
+            # a complex-valued function of a real Herm/Sym is always `Symmetric`
+            Symmetric(fA)
+        else
+            # Julia 1.12 made real-valued matrix functions of `Hermitian{<:Real}`
+            # return `Hermitian` (they previously returned `Symmetric`); match the
+            # type of the primal `f(A)`.
+            @static if VERSION >= v"1.12"
+                _symhermtype(A)(fA)
+            else
+                Symmetric(fA)
+            end
+        end
     elseif eltype(fλ) <: Complex
         fA
     else
diff --git a/test/rulesets/LinearAlgebra/symmetric.jl b/test/rulesets/LinearAlgebra/symmetric.jl
index 593b82148..1940ca15e 100644
--- a/test/rulesets/LinearAlgebra/symmetric.jl
+++ b/test/rulesets/LinearAlgebra/symmetric.jl
@@ -328,8 +328,11 @@
                     if is_inferable(f, A)
                         Y_ad, ∂Y_ad = @maybe_inferred frule((ZeroTangent(), ΔA), f, A)
                     else
+                        # real-valued functions of `TA{<:Real}` return `TA` on Julia
+                        # 1.12+ (they previously always returned `Symmetric`)
+                        SR = @static VERSION >= v"1.12" ? TA{T} : Symmetric{T}
                         TY = T∂Y = if T <: Real
-                            Union{Symmetric{Complex{T}},Symmetric{T}}
+                            Union{Symmetric{Complex{T}},SR}
                         else
                             Union{Matrix{T},Hermitian{T}}
                         end
@@ -381,8 +384,11 @@
                     if is_inferable(f, A)
                         Y_ad, back = @maybe_inferred rrule(f, A)
                     else
+                        # real-valued functions of `TA{<:Real}` return `TA` on Julia
+                        # 1.12+ (they previously always returned `Symmetric`)
+                        SR = @static VERSION >= v"1.12" ? TA{T} : Symmetric{T}
                         TY = if T <: Real
-                            Union{Symmetric{Complex{T}},Symmetric{T}}
+                            Union{Symmetric{Complex{T}},SR}
                         else
                             Union{Matrix{T},Hermitian{T}}
                         end