diff --git a/lib/ModelingToolkitTearing/Project.toml b/lib/ModelingToolkitTearing/Project.toml
index 8397003..8f54e43 100644
--- a/lib/ModelingToolkitTearing/Project.toml
+++ b/lib/ModelingToolkitTearing/Project.toml
@@ -36,7 +36,7 @@ OrderedCollections = "1.8.1"
 SciMLBase = "2.108"
 Setfield = "0.7, 0.8, 1"
 SparseArrays = "1"
-StateSelection = "1.8"
+StateSelection = "1.9"
 SymbolicIndexingInterface = "0.3"
 SymbolicUtils = "4.3"
 Symbolics = "7.15.1"
diff --git a/lib/ModelingToolkitTearing/src/clock_inference/interface.jl b/lib/ModelingToolkitTearing/src/clock_inference/interface.jl
index 6ab5e59..1ae4916 100644
--- a/lib/ModelingToolkitTearing/src/clock_inference/interface.jl
+++ b/lib/ModelingToolkitTearing/src/clock_inference/interface.jl
@@ -69,6 +69,7 @@ function system_subset(ts::TearingState, ieqs::Vector{Int}, iieqs::Vector{Int},
     @set! ts.sys.initialization_eqs = initeqs[iieqs]
     @set! ts.original_eqs = ts.original_eqs[ieqs]
     @set! ts.structure = system_subset(ts.structure, ieqs, ivars)
+    @set! ts.always_present = ts.always_present[ivars]
     if !isempty(ts.eqs_source)
         @set! ts.eqs_source = ts.eqs_source[ieqs]
     end
diff --git a/lib/ModelingToolkitTearing/src/stateselection_interface.jl b/lib/ModelingToolkitTearing/src/stateselection_interface.jl
index b079e0f..f092d77 100644
--- a/lib/ModelingToolkitTearing/src/stateselection_interface.jl
+++ b/lib/ModelingToolkitTearing/src/stateselection_interface.jl
@@ -5,6 +5,7 @@ function StateSelection.var_derivative!(ts::TearingState, v::Int)
     D = Differential(MTKBase.get_iv(sys))
     push!(ts.fullvars, D(ts.fullvars[v]))
     push!(ts.structure.state_priorities, ts.structure.state_priorities[v])
+    push!(ts.always_present, ts.always_present[v])
     return var_diff
 end
 
@@ -348,6 +349,7 @@ function StateSelection.rm_eqs_vars!(
         structure, eqs_to_rm, vars_to_rm; eqs_sorted_and_uniqued, vars_sorted_and_uniqued
     )
     deleteat!(state.fullvars, vars_to_rm)
+    deleteat!(state.always_present, vars_to_rm)
     eqs = copy(MTKBase.get_eqs(state.sys))
     deleteat!(eqs, eqs_to_rm)
     deleteat!(state.original_eqs, eqs_to_rm)
@@ -361,3 +363,6 @@ function StateSelection.rm_eqs_vars!(
     return old_to_new_eq, old_to_new_var
 end
 
+function StateSelection.is_unused_var(state::TearingState, var::Integer)
+    return !state.always_present[var] && isempty(𝑑neighbors(state.structure.graph, var))
+end
diff --git a/lib/ModelingToolkitTearing/src/tearingstate.jl b/lib/ModelingToolkitTearing/src/tearingstate.jl
index 6088642..bc80f50 100644
--- a/lib/ModelingToolkitTearing/src/tearingstate.jl
+++ b/lib/ModelingToolkitTearing/src/tearingstate.jl
@@ -59,6 +59,23 @@ mutable struct TearingState <: StateSelection.TransformationState{System}
     are not used in the rest of the system.
     """
     additional_observed::Vector{Equation}
+    """
+    Corresponding to `fullvars`, marks variables which may not be structurally present in
+    any equation according to `structure.graph` but should not be considered as unused.
+    This is typically used by variables on the RHS of equations in `additional_observed`,
+    and is useful for ensuring the consistency check is valid. For example, a simplification
+    pass prior to `StateSelection.check_consistency` may process the equations
+    ```julia
+    a ~ b
+    b ~ c
+    c ~ a
+    ```
+    Into `additional_observed = [a ~ b, c ~ b]`, and thus end up in a state where
+    there are no equations and `fullvars = [b]`. The consistency check would consider `b` as
+    unused and the system as fully determined, but in reality `b` should be considered
+    as used and the system singular.
+    """
+    always_present::BitVector
     statemachines::Vector{System}
     """
     Source information for each equation in the `TearingState`. `Vector{Symbol}` for each
@@ -325,7 +342,8 @@ function TearingState(sys::System, source_info::Union{Nothing, MTKBase.EquationS
     structure = SystemStructure(complete(var_to_diff), complete(eq_to_diff),
                                 complete(graph), nothing, var_types, state_priorities, false)
     return TearingState(sys, fullvars, structure, Equation[], param_derivative_map,
-                        no_deriv_params, original_eqs, Equation[], typeof(sys)[], sources)
+                        no_deriv_params, original_eqs, Equation[], falses(length(fullvars)),
+                        typeof(sys)[], sources)
 end
 
 function build_state_priorities(sys::System, fullvars::Vector{SymbolicT}, var_to_diff::StateSelection.DiffGraph)