diff --git a/.github/workflows/JET.yml b/.github/workflows/JET.yml
index 3de143d..53e46bc 100644
--- a/.github/workflows/JET.yml
+++ b/.github/workflows/JET.yml
@@ -48,8 +48,6 @@ jobs:
           julia --project=. --color=yes -e '
             using JET, Test
             using FixedPointNumbers
-            println("report_package(FixedPointNumbers, toplevel_logger=nothing)")
-            show(JET.report_package(FixedPointNumbers, toplevel_logger=nothing))
             @testset "JET" begin
-                JET.test_package(FixedPointNumbers; target_modules=(FixedPointNumbers,))
+                JET.test_package(FixedPointNumbers)
             end'
diff --git a/src/FixedPointNumbers.jl b/src/FixedPointNumbers.jl
index 7bafaca..0b91cf7 100644
--- a/src/FixedPointNumbers.jl
+++ b/src/FixedPointNumbers.jl
@@ -67,6 +67,17 @@ wrapping_rem(x::Real, ::Type{X}) where {X <: FixedPoint} = _rem(x, X)
 saturating_rem(x::Real, ::Type{X}) where {X <: FixedPoint} = _rem(x, X)
 checked_rem(x::Real, ::Type{X}) where {X <: FixedPoint} = _rem(x, X)
 
+# `Type{X} where {X <: FixedPoint}` also matches `Type{Union{}}`. Since Julia 1.10, a method of the form
+# `f(..., ::Type{Union{}}, slurp...)` takes precedence over all other methods of `f`, so these calls throw a
+# plain `MethodError` instead of an ambiguity error or an error from inside another method.
+wrapping_mul(x, ::Type{Union{}}, slurp...) = throw(MethodError(wrapping_mul, (x, Union{}, slurp...)))
+saturating_mul(x, ::Type{Union{}}, slurp...) = throw(MethodError(saturating_mul, (x, Union{}, slurp...)))
+wrapping_rem(x, ::Type{Union{}}, slurp...) = throw(MethodError(wrapping_rem, (x, Union{}, slurp...)))
+saturating_rem(x, ::Type{Union{}}, slurp...) = throw(MethodError(saturating_rem, (x, Union{}, slurp...)))
+# `*`, `rem` and the functions of `Base.Checked` belong to Base, so the internal functions they call are guarded
+_convert(::Type{Union{}}, slurp...) = throw(MethodError(convert, (Union{}, slurp...)))
+_rem(x, ::Type{Union{}}, slurp...) = throw(MethodError(rem, (x, Union{}, slurp...)))
+
 # constructor-style conversions
 (::Type{X})(x::X) where {X <: FixedPoint}      = x
 (::Type{X})(x::Number) where {X <: FixedPoint} = _convert(X, x)
@@ -204,6 +215,7 @@ floattype(::Type{X}) where {T <: LongInts, X <: FixedPoint{T}} = BigFloat
 floattype(::Type{Complex{T}}) where T = Complex{floattype(T)}
 floattype(::Type{Base.TwicePrecision{Float64}}) = Float64    # wider would be nice, but hardware support is paramount
 floattype(::Type{Base.TwicePrecision{T}}) where T<:Union{Float16,Float32} = widen(T)
+floattype(::Type{Union{}}, slurp...) = throw(MethodError(floattype, (Union{}, slurp...)))
 
 float(x::FixedPoint) = convert(floattype(x), x)
 
@@ -363,13 +375,10 @@ clamp(x::X, lo::X, hi::X) where {X <: FixedPoint} = X(clamp(x.i, lo.i, hi.i), 0)
 
 clamp(x, ::Type{X}) where {X <: FixedPoint} = clamp(x, typemin(X), typemax(X)) % X
 
-# Since `FixedPoint` is not an integer type, it is not clear in what type
-# `signed` and `unsigned` for `FixedPoint` should return values. They should
-# currently throw errors in case we support "unsigned Fixed" or "signed Normed"
-# in the future. The following "incomplete" code is necessary for Julia v1.0
-# etc. to prevent accidental conversion to an integer type.
-signed(x::X) where {X <: FixedPoint} = signed(X)(signed(x.i), 0)
-unsigned(x::X) where {X <: FixedPoint} = unsigned(X)(unsigned(x.i), 0)
+# Throw a `MethodError` so that the generic fallbacks don't convert a `FixedPoint`
+# to an integer type.
+signed(x::FixedPoint) = throw(MethodError(signed, (x,)))
+unsigned(x::FixedPoint) = throw(MethodError(unsigned, (x,)))
 
 function sign(x::X) where {T, X <: FixedPoint{T}}
     T <: Unsigned && return iszero(x) ? x : oneunit(X)
@@ -614,6 +623,7 @@ scaledual(::Type{Tdual}, x::Union{Number,AbstractArray{<:Number}}) where Tdual =
 scaledual(::Type{Tdual}, x::FixedPoint) where Tdual = convert(Tdual, 1/rawone(x)), reinterpret(x)
 scaledual(::Type{Tdual}, x::AbstractArray{T}) where {Tdual, T <: FixedPoint} =
     convert(Tdual, 1/rawone(T)), reinterpret(rawtype(T), x)
+scaledual(::Type{Union{}}, slurp...) = throw(MethodError(scaledual, (Union{}, slurp...)))
 
 @noinline function throw_converterror(::Type{X}, @nospecialize(x)) where X <: FixedPoint
     nbits = bitwidth(rawtype(X))
@@ -657,7 +667,7 @@ function Random.rand(r::AbstractRNG, ::SamplerType{X}) where X <: FixedPoint
 end
 
 function rand!(r::AbstractRNG, A::Array{X}, ::SamplerType{X}) where {T, X <: FixedPoint{T}}
-    At = unsafe_wrap(Array, reinterpret(Ptr{T}, pointer(A)), size(A))
+    At = unsafe_wrap(Array, Ptr{T}(pointer(A)), size(A))
     Random.rand!(r, At, SamplerType{T}())
     A
 end
diff --git a/src/fixed.jl b/src/fixed.jl
index 212a95f..ce931c4 100644
--- a/src/fixed.jl
+++ b/src/fixed.jl
@@ -189,10 +189,13 @@ function _trunc(x::Fixed{T,f}) where {T, f}
     _rawone = oneunit(T) << f
     reinterpret(Fixed{T,f}, (x.i < 0) & (r != 0) ? t + _rawone : t)
 end
-function _floor(x::Fixed{T,f}) where {T, f}
+# The bound `T <: Signed` limits the overlap with the generic `_trunc` for `FixedPoint{<:Unsigned}` to
+# `Fixed{Union{}, f}`, which has no values and gets its own method.
+function _floor(x::Fixed{T,f}) where {T <: Signed, f}
     f == bitwidth(T) && x.i < 0 && throw_converterror(Fixed{T,f}, -1) # TODO: remove this line
     Fixed{T,f}(x.i & intmask(x), 0)
 end
+_floor(x::Fixed{Union{}, f}) where f = throw(MethodError(_floor, (x,))) # COV_EXCL_LINE
 function _ceil(x::Fixed{T,f}) where {T, f}
     f == 0 && return x
     upper = typemax(T) & intmask(x)
diff --git a/test/traits.jl b/test/traits.jl
index c248b4c..0d135a4 100644
--- a/test/traits.jl
+++ b/test/traits.jl
@@ -18,3 +18,14 @@ struct MyReal <: Real end
 
     @test_skip(@test_throws MethodError floattype(MyReal))   # TODO: eliminate `@test_skipped` when depwarn is eliminated. See #177.
 end
+
+@testset "`Union{}` as type argument" begin
+    @test_throws MethodError floattype(Union{})
+    @test_throws MethodError scaledual(Union{}, 0.5N0f8)
+    @test_throws MethodError 0.5 * Union{}
+    @test_throws MethodError 0.5 % Union{}
+    for f in (wrapping_mul, saturating_mul, Base.Checked.checked_mul,
+              wrapping_rem, saturating_rem, Base.Checked.checked_rem)
+        @test_throws MethodError f(0.5, Union{})
+    end
+end