diff --git a/stdlib/LinearAlgebra/src/special.jl b/stdlib/LinearAlgebra/src/special.jl
index 5a7c98cfdf32c..32a5476842933 100644
--- a/stdlib/LinearAlgebra/src/special.jl
+++ b/stdlib/LinearAlgebra/src/special.jl
@@ -112,6 +112,8 @@ for op in (:+, :-)
     end
 end
 
+(*)(Da::Diagonal, A::BandedMatrix, Db::Diagonal) = _tri_matmul(Da, A, Db)
+
 # disambiguation between triangular and banded matrices, banded ones "dominate"
 _mul!(C::AbstractMatrix, A::AbstractTriangular, B::BandedMatrix, alpha::Number, beta::Number) =
     @stable_muladdmul _mul!(C, A, B, MulAddMul(alpha, beta))
diff --git a/stdlib/LinearAlgebra/test/diagonal.jl b/stdlib/LinearAlgebra/test/diagonal.jl
index dfb901908ba69..98f5498c71033 100644
--- a/stdlib/LinearAlgebra/test/diagonal.jl
+++ b/stdlib/LinearAlgebra/test/diagonal.jl
@@ -1265,6 +1265,17 @@ end
     @test *(Diagonal(ones(n)), Diagonal(1:n), Diagonal(ones(n)), Diagonal(1:n)) isa Diagonal
 end
 
+@testset "triple multiplication with a sandwiched BandedMatrix" begin
+    D = Diagonal(StepRangeLen(NaN, 0, 4));
+    B = Bidiagonal(1:4, 1:3, :U)
+    C = D * B * D
+    @test iszero(diag(C, 2))
+    # test associativity
+    C1 = (D * B) * D
+    C2 = D * (B * D)
+    @test diag(C,2) == diag(C1,2) == diag(C2,2)
+end
+
 @testset "diagind" begin
     D = Diagonal(1:4)
     M = Matrix(D)