diff --git a/src/derivatives/broadcast.jl b/src/derivatives/broadcast.jl
index cacad7e..3632dfc 100644
--- a/src/derivatives/broadcast.jl
+++ b/src/derivatives/broadcast.jl
@@ -38,8 +38,14 @@ struct TrackedStyle{N} <: AbstractArrayStyle{N} end
 (::Type{<:TrackedStyle})(::Val{N}) where {N} = TrackedStyle{N}()
 
 Broadcast.BroadcastStyle(::Type{<:TrackedReal}) = TrackedStyle{0}()
-# only an element type that fixes `D`, since `getouttype` reads it off
-Broadcast.BroadcastStyle(::Type{<:AbstractArray{<:TrackedReal{<:Any, D}, N}}) where {D, N} =
+# invariant element types that fix `D`, so that `Union{}` (a subtype of every type) does not match
+Broadcast.BroadcastStyle(::Type{<:AbstractArray{TrackedReal{V, D, O}, N}}) where {V, D, O, N} =
+    TrackedStyle{N}()
+Broadcast.BroadcastStyle(::Type{<:AbstractArray{TrackedReal{V, D, O} where {O}, N}}) where {V, D, N} =
+    TrackedStyle{N}()
+Broadcast.BroadcastStyle(::Type{<:AbstractArray{TrackedReal{V, D, O} where {V <: Real}, N}}) where {D, O, N} =
+    TrackedStyle{N}()
+Broadcast.BroadcastStyle(::Type{<:AbstractArray{TrackedReal{V, D, O} where {V <: Real, O}, N}}) where {D, N} =
     TrackedStyle{N}()
 
 # `AbstractArrayStyle{Any}` carries `Any` as its dimension, which `max` cannot compare
@@ -101,6 +107,8 @@ end
 
 getouttype(::TrackedReal{<:Any, D}) where {D} = D
 getouttype(::AbstractArray{<:TrackedReal{<:Any, D}}) where {D} = D
+# `Union{} <: TrackedReal`, but binds no `D`
+getouttype(::AbstractArray{Union{}}) = Union{}
 getouttype(::Any) = Union{}
 
 # a perturbation riding on an argument ends up in the derivative, so `D` has to hold it. An
diff --git a/test/derivatives/BroadcastTests.jl b/test/derivatives/BroadcastTests.jl
index 705aec7..22fe8f4 100644
--- a/test/derivatives/BroadcastTests.jl
+++ b/test/derivatives/BroadcastTests.jl
@@ -56,6 +56,21 @@ end
     # `Base` styles scalars as `AbstractArrayStyle{0}`
     @test BroadcastStyle(typeof(track(rand(), tp))) === TrackedStyle{0}()
     @test TrackedStyle{1}(Val(2)) === TrackedStyle{2}()
+    # element types that leave the value type or the origin free
+    @test BroadcastStyle(Vector{TrackedReal{Float64, Float64}}) === TrackedStyle{1}()
+    @test BroadcastStyle(Vector{TrackedReal{<:Real, Float64}}) === TrackedStyle{1}()
+    # `TrackedReal{<:Real, Float64, Nothing}`, as built by a comprehension over mixed value types
+    a = TrackedReal(1.0, 0.0)
+    b = TrackedReal(ForwardDiff.Dual(1.0, 1.0), 0.0)
+    @test BroadcastStyle(typeof([x for x in Real[a, b]])) === TrackedStyle{1}()
+end
+
+@testset "arrays that hold no element keep `Base`'s style (#324)" begin
+    # `Union{} <: TrackedReal`
+    @test BroadcastStyle(Vector{Union{}}) === DefaultArrayStyle{1}()
+    @test BroadcastStyle(Matrix{Union{}}) === DefaultArrayStyle{2}()
+    @test String.(Union{}[]) isa Vector{Union{}}
+    @test isempty(track(1.0, InstructionTape()) .+ Union{}[])
 end
 
 @testset "`BroadcastStyle` precedence" begin