diff --git a/ext/MathOptComplementsJuMPExt.jl b/ext/MathOptComplementsJuMPExt.jl
index f645730..1d959a1 100644
--- a/ext/MathOptComplementsJuMPExt.jl
+++ b/ext/MathOptComplementsJuMPExt.jl
@@ -13,16 +13,8 @@ function MathOptComplements.Bridges.add_all_bridges(
     model::JuMP.GenericModel{T},
     ::Type{U} = T,
 ) where {T,U}
-    for BT in (
-        MathOptComplements.Bridges.SpecifySetTypeBridge,
-        MathOptComplements.Bridges.ComplementsVectorizeBridge,
-        MathOptComplements.Bridges.SplitIntervalBridge,
-        MathOptComplements.Bridges.FlipSignBridge,
-        MathOptComplements.Bridges.ToSOS1Bridge,
-        MathOptComplements.Bridges.VerticalBridge,
-        MathOptComplements.Bridges.NonlinearBridge,
-    )
-        JuMP.add_bridge(model, BT; coefficient_type = U)
+    for bridge_type in MathOptComplements.Bridges._ALL_BRIDGE_TYPES
+        JuMP.add_bridge(model, bridge_type; coefficient_type = U)
     end
     return
 end
diff --git a/src/Bridges/Bridges.jl b/src/Bridges/Bridges.jl
index 5d1d9c7..c300648 100644
--- a/src/Bridges/Bridges.jl
+++ b/src/Bridges/Bridges.jl
@@ -21,6 +21,17 @@ include("NonlinearBridge.jl")
 include("ToSOS1Bridge.jl")
 include("ToZerosBridge.jl")
 
+const _ALL_BRIDGE_TYPES = Any[
+    SpecifySetTypeBridge,
+    ComplementsVectorizeBridge,
+    SplitIntervalBridge,
+    FlipSignBridge,
+    ToSOS1Bridge,
+    ToZerosBridge,
+    VerticalBridge,
+    NonlinearBridge,
+]
+
 """
     add_all_bridges(model::MOI.ModelLike, ::Type{T} = Float64)
 
@@ -36,14 +47,9 @@ the default [`ScholtesRelaxation`](@ref) because the
 attribute is only supported by [`MathOptComplements.Optimizer`](@ref).
 """
 function add_all_bridges(model::MOI.ModelLike, ::Type{T} = Float64) where {T}
-    MOI.Bridges.add_bridge(model, SpecifySetTypeBridge{T})
-    MOI.Bridges.add_bridge(model, ComplementsVectorizeBridge{T})
-    MOI.Bridges.add_bridge(model, SplitIntervalBridge{T})
-    MOI.Bridges.add_bridge(model, FlipSignBridge{T})
-    MOI.Bridges.add_bridge(model, ToSOS1Bridge{T})
-    MOI.Bridges.add_bridge(model, ToZerosBridge{T})
-    MOI.Bridges.add_bridge(model, VerticalBridge{T})
-    MOI.Bridges.add_bridge(model, NonlinearBridge{T})
+    for bridge_type in _ALL_BRIDGE_TYPES
+        MOI.Bridges.add_bridge(model, bridge_type{T})
+    end
     return
 end
 
diff --git a/test/Bridges/lazy.jl b/test/Bridges/lazy.jl
new file mode 100644
index 0000000..d66cef8
--- /dev/null
+++ b/test/Bridges/lazy.jl
@@ -0,0 +1,134 @@
+# Copyright (c) 2025 François Pacaud, Benoît Legat, and contributors
+#
+# Use of this source code is governed by an MIT-style license that can be found
+# in the LICENSE.md file or at https://opensource.org/licenses/MIT.
+
+# Integration tests for the bridge selection performed by a
+# `MOI.Bridges.LazyBridgeOptimizer` set up with `add_all_bridges` (as opposed to
+# `MathOptComplements.Optimizer`, which selects the bridge explicitly). The
+# `LazyBridgeOptimizer` picks a bridge based on the constraint types each bridge
+# declares it adds (`added_constraint_types`) and whether they are supported by
+# the solver. In particular, for a MIP solver (which supports neither quadratic
+# nor nonlinear constraints), the `NonlinearBridge` must *not* be selected.
+
+module TestLazyBridge
+
+using Test
+
+import MathOptComplements
+import MathOptInterface as MOI
+
+const MOC = MathOptComplements
+
+# A MIP solver: it supports integrality and linear constraints (including
+# `SOS1` only through the `SOS1ToMILP` bridge), but neither quadratic nor
+# nonlinear constraints.
+MOI.Utilities.@model(
+    MIPModel,
+    (MOI.ZeroOne, MOI.Integer),
+    (MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval),
+    (MOI.Nonnegatives, MOI.Nonpositives, MOI.Zeros),
+    (),
+    (),
+    (MOI.ScalarAffineFunction,),
+    (MOI.VectorOfVariables,),
+    (MOI.VectorAffineFunction,),
+)
+
+function MOI.supports_constraint(
+    ::MIPModel,
+    ::Type{MOI.ScalarNonlinearFunction},
+    ::Type{<:MOI.AbstractScalarSet},
+)
+    return false
+end
+
+# An NLP solver: it supports quadratic and nonlinear constraints but not `SOS1`.
+MOI.Utilities.@model(
+    NLPModel,
+    (MOI.ZeroOne, MOI.Integer),
+    (MOI.EqualTo, MOI.GreaterThan, MOI.LessThan, MOI.Interval),
+    (MOI.Nonnegatives, MOI.Nonpositives, MOI.Zeros),
+    (),
+    (),
+    (MOI.ScalarAffineFunction, MOI.ScalarQuadraticFunction),
+    (MOI.VectorOfVariables,),
+    (MOI.VectorAffineFunction,),
+)
+
+function MOI.supports_constraint(
+    ::NLPModel,
+    ::Type{MOI.ScalarNonlinearFunction},
+    ::Type{<:MOI.AbstractScalarSet},
+)
+    return true
+end
+
+function runtests()
+    is_test(name) = startswith("$name", "test_")
+    @testset "$name" for name in filter(is_test, names(@__MODULE__; all = true))
+        getfield(@__MODULE__, name)()
+    end
+    return
+end
+
+function _lazy(inner)
+    lazy = MOI.Bridges.full_bridge_optimizer(inner, Float64)
+    MOC.Bridges.add_all_bridges(lazy, Float64)
+    return lazy
+end
+
+const _SET_TYPES = (
+    MOI.Nonnegatives,
+    MOI.Nonpositives,
+    MOI.GreaterThan{Float64},
+    MOI.LessThan{Float64},
+    MOI.Interval{Float64},
+    MOI.Zeros,
+)
+
+function test_mip_does_not_select_nonlinear_bridge()
+    lazy = _lazy(MIPModel{Float64}())
+    # The `MOI.Complements` constraint is bridgeable: this requires the whole
+    # SOS1/MILP chain, including `ToZerosBridge` for the `Zeros` case produced
+    # by `SpecifySetTypeBridge`.
+    @test MOI.supports_constraint(lazy, MOI.VectorOfVariables, MOI.Complements)
+    for Sset in _SET_TYPES
+        S = MOC.ComplementsWithSetType{Sset}
+        @test MOI.supports_constraint(lazy, MOI.VectorOfVariables, S)
+        bt = MOI.Bridges.bridge_type(lazy, MOI.VectorOfVariables, S)
+        @test !(bt <: MOC.Bridges.NonlinearBridge)
+    end
+    # The two terminal bridges of the MILP path.
+    @test MOI.Bridges.bridge_type(
+        lazy,
+        MOI.VectorOfVariables,
+        MOC.ComplementsWithSetType{MOI.Nonnegatives},
+    ) <: MOC.Bridges.ToSOS1Bridge
+    @test MOI.Bridges.bridge_type(
+        lazy,
+        MOI.VectorOfVariables,
+        MOC.ComplementsWithSetType{MOI.Zeros},
+    ) <: MOC.Bridges.ToZerosBridge
+    return
+end
+
+function test_nlp_selects_nonlinear_bridge()
+    lazy = _lazy(NLPModel{Float64}())
+    # When the solver supports nonlinear constraints, `NonlinearBridge` is
+    # selected (it is cheaper than the SOS1/MILP chain since the solver does
+    # not support `SOS1` natively).
+    for Sset in (MOI.Nonnegatives, MOI.Interval{Float64})
+        bt = MOI.Bridges.bridge_type(
+            lazy,
+            MOI.VectorOfVariables,
+            MOC.ComplementsWithSetType{Sset},
+        )
+        @test bt <: MOC.Bridges.NonlinearBridge
+    end
+    return
+end
+
+end  # TestLazyBridge
+
+TestLazyBridge.runtests()
diff --git a/test/runtests.jl b/test/runtests.jl
index 72e0982..86a9b3f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -12,3 +12,9 @@ is_test(f) = startswith(f, "test_") && endswith(f, ".jl")
         include(joinpath(root, file))
     end
 end
+
+# Integration test that is not named `test_*.jl` so that it is not picked up by
+# the `walkdir` above; it is included explicitly here.
+@testset "Bridges/lazy.jl" begin
+    include(joinpath(@__DIR__, "Bridges", "lazy.jl"))
+end