diff --git a/src/diagonal.jl b/src/diagonal.jl
index 9f8d54e5..aec54708 100644
--- a/src/diagonal.jl
+++ b/src/diagonal.jl
@@ -671,22 +671,24 @@ end
 for Tri in (:UpperTriangular, :LowerTriangular)
     UTri = Symbol(:Unit, Tri)
     # 2 args
-    for (fun, f) in zip((:*, :rmul!, :rdiv!, :/), (:identity, :identity, :inv, :inv))
-        @eval $fun(A::$Tri, D::Diagonal) = $Tri($fun(A.data, D))
-        @eval $fun(A::$UTri, D::Diagonal) = $Tri(_setdiag!($fun(A.data, D), $f, D.diag))
+    for (fun, f) in zip((:mul, :rmul!, :rdiv!, :/), (:identity, :identity, :inv, :inv))
+        g = fun == :mul ? :* : fun
+        @eval $fun(A::$Tri, D::Diagonal) = $Tri($g(A.data, D))
+        @eval $fun(A::$UTri, D::Diagonal) = $Tri(_setdiag!($g(A.data, D), $f, D.diag))
     end
-    @eval *(A::$Tri{<:Any, <:StridedMaybeAdjOrTransMat}, D::Diagonal) =
-            @invoke *(A::AbstractMatrix, D::Diagonal)
-    @eval *(A::$UTri{<:Any, <:StridedMaybeAdjOrTransMat}, D::Diagonal) =
-            @invoke *(A::AbstractMatrix, D::Diagonal)
-    for (fun, f) in zip((:*, :lmul!, :ldiv!, :\), (:identity, :identity, :inv, :inv))
-        @eval $fun(D::Diagonal, A::$Tri) = $Tri($fun(D, A.data))
-        @eval $fun(D::Diagonal, A::$UTri) = $Tri(_setdiag!($fun(D, A.data), $f, D.diag))
+    @eval mul(A::$Tri{<:Any, <:StridedMaybeAdjOrTransMat}, D::Diagonal) =
+            @invoke mul(A::AbstractMatrix, D::Diagonal)
+    @eval mul(A::$UTri{<:Any, <:StridedMaybeAdjOrTransMat}, D::Diagonal) =
+            @invoke mul(A::AbstractMatrix, D::Diagonal)
+    for (fun, f) in zip((:mul, :lmul!, :ldiv!, :\), (:identity, :identity, :inv, :inv))
+        g = fun == :mul ? :* : fun
+        @eval $fun(D::Diagonal, A::$Tri) = $Tri($g(D, A.data))
+        @eval $fun(D::Diagonal, A::$UTri) = $Tri(_setdiag!($g(D, A.data), $f, D.diag))
     end
-    @eval *(D::Diagonal, A::$Tri{<:Any, <:StridedMaybeAdjOrTransMat}) =
-            @invoke *(D::Diagonal, A::AbstractMatrix)
-    @eval *(D::Diagonal, A::$UTri{<:Any, <:StridedMaybeAdjOrTransMat}) =
-            @invoke *(D::Diagonal, A::AbstractMatrix)
+    @eval mul(D::Diagonal, A::$Tri{<:Any, <:StridedMaybeAdjOrTransMat}) =
+            @invoke mul(D::Diagonal, A::AbstractMatrix)
+    @eval mul(D::Diagonal, A::$UTri{<:Any, <:StridedMaybeAdjOrTransMat}) =
+            @invoke mul(D::Diagonal, A::AbstractMatrix)
     # 3-arg ldiv!
     @eval ldiv!(C::$Tri, D::Diagonal, A::$Tri) = $Tri(ldiv!(C.data, D, A.data))
     @eval ldiv!(C::$Tri, D::Diagonal, A::$UTri) = $Tri(_setdiag!(ldiv!(C.data, D, A.data), inv, D.diag))
diff --git a/src/matmul.jl b/src/matmul.jl
index 21c6e299..84a83d3a 100644
--- a/src/matmul.jl
+++ b/src/matmul.jl
@@ -111,7 +111,9 @@ julia> [1 1; 0 1] * [1 0; 1 1]
  1  1
 ```
 """
-function (*)(A::AbstractMatrix, B::AbstractMatrix)
+(*)(A::AbstractMatrix, B::AbstractMatrix) = mul(A, B)
+# we add an extra level of indirection to avoid ambiguities in *
+function mul(A::AbstractMatrix, B::AbstractMatrix)
     TS = promote_op(matprod, eltype(A), eltype(B))
     mul!(matprod_dest(A, B, TS), A, B)
 end