diff --git a/src/MOI_wrapper.jl b/src/MOI_wrapper.jl
index f5f382a..eb29d8a 100644
--- a/src/MOI_wrapper.jl
+++ b/src/MOI_wrapper.jl
@@ -70,6 +70,18 @@ function MOI.Bridges.supports_bridging_constrained_variable(
     return true
 end
 
+# The variables are not bridged (`is_variable_bridged` is `false`), so
+# `add_constrained_variables` falls back to adding the variables and then the
+# `MOI.VectorOfVariables` constraint. This is only needed for
+# `MOI.VariableBridgingCost`, which `MOI.Utilities.copy_to` queries to decide in
+# which order the variables are added.
+function MOI.Bridges.bridge_type(
+    b::Optimizer,
+    S::Type{<:ComplementsWithSetType},
+)
+    return MOI.Bridges.bridge_type(b, MOI.VectorOfVariables, S)
+end
+
 # No objective bridge
 MOI.Bridges.is_bridged(::Optimizer, ::Type{<:MOI.AbstractFunction}) = false
 
diff --git a/test/test_optimizer.jl b/test/test_optimizer.jl
index 196e28b..3218160 100644
--- a/test/test_optimizer.jl
+++ b/test/test_optimizer.jl
@@ -522,6 +522,64 @@ function test_interval_complements_solve_HiGHS()
     return
 end
 
+function test_ComplementsWithSetType_added_directly_HiGHS()
+    # The user gives the set type directly instead of going through
+    # `MOI.Complements`/`SpecifySetTypeBridge`. Because the shift by the lower
+    # bound of `x2` is now computed in `final_touch`, the constraint can even be
+    # added *before* that bound is set.
+    model = Model(
+        () -> MathOptComplements.Optimizer(
+            MOI.instantiate(HiGHS.Optimizer; with_bridge_type = Float64),
+        ),
+    )
+    set_silent(model)
+    @variable(model, x1)
+    @variable(model, x2)
+    S = MathOptComplements.ComplementsWithSetType{MOI.GreaterThan{Float64}}
+    @constraint(model, [x1, x2] in S(2))
+    # Finite bounds are needed by `SOS1ToMILPBridge` (HiGHS has neither native
+    # SOS1 nor nonlinear support).
+    set_lower_bound(x1, 0.0)
+    set_upper_bound(x1, 1e3)
+    set_lower_bound(x2, 1.0)
+    set_upper_bound(x2, 1e3)
+    @constraint(model, x1 + x2 >= 2)
+    @objective(model, Min, x1 + 3 * x2)
+    optimize!(model)
+    @test is_solved_and_feasible(model)
+    # The complementarity is `x1 ⟂ (x2 - 1)`, so either `x2 = 1` and `x1 >= 1`
+    # (objective `4`), or `x1 = 0` and `x2 >= 2` (objective `6`).
+    @test isapprox(objective_value(model), 4.0; atol = 1e-6)
+    @test isapprox(value(x1), 1.0; atol = 1e-6)
+    @test isapprox(value(x2), 1.0; atol = 1e-6)
+    return
+end
+
+function test_ComplementsWithSetType_added_directly_Ipopt()
+    # Same model as above but the inner solver supports nonlinear constraints,
+    # so `NonlinearBridge` reformulates the complementarity instead.
+    model = Model(
+        () -> MathOptComplements.Optimizer(
+            MOI.instantiate(Ipopt.Optimizer; with_cache_type = Float64),
+        ),
+    )
+    set_silent(model)
+    @variable(model, x1)
+    @variable(model, x2)
+    S = MathOptComplements.ComplementsWithSetType{MOI.GreaterThan{Float64}}
+    @constraint(model, [x1, x2] in S(2))
+    set_lower_bound(x1, 0.0)
+    set_lower_bound(x2, 1.0)
+    @constraint(model, x1 + x2 >= 2)
+    @objective(model, Min, x1 + 3 * x2)
+    optimize!(model)
+    @test is_solved_and_feasible(model)
+    @test isapprox(objective_value(model), 4.0; atol = 1e-6)
+    @test isapprox(value(x1), 1.0; atol = 1e-6)
+    @test isapprox(value(x2), 1.0; atol = 1e-6)
+    return
+end
+
 function test_add_all_bridges_JuMP_GenericModel()
     model = Model(Ipopt.Optimizer)
     MathOptComplements.Bridges.add_all_bridges(model)