diff --git a/src/timesteppers/heun.jl b/src/timesteppers/heun.jl
index 0dfd4dee3..7a1546813 100644
--- a/src/timesteppers/heun.jl
+++ b/src/timesteppers/heun.jl
@@ -31,11 +31,11 @@ struct HeunCache{NF, P, T, NS} <: AbstractTimeStepperCache{NF}
     namespaces::NS
 end
 
-# Allocate the Heun cache by deep-copying the prognostic and tendency field containers, recursing into
+# Allocate the Heun cache with a `similar` field for each prognostic and tendency field, recursing into
 # each namespace so the cache tree mirrors the state tree.
 function initialize(heun::Heun, state::AbstractStateVariables)
-    prognostic = map(deepcopy, getfield(state, :prognostic))
-    tendencies = map(deepcopy, getfield(state, :tendencies))
+    prognostic = map(similar, getfield(state, :prognostic))
+    tendencies = map(similar, getfield(state, :tendencies))
     namespaces = map(ns -> initialize(heun, ns), getfield(state, :namespaces))
     return HeunCache{eltype(state), typeof(prognostic), typeof(tendencies), typeof(namespaces)}(prognostic, tendencies, namespaces)
 end
diff --git a/test/timestepping/heun.jl b/test/timestepping/heun.jl
index 129502abf..f1e919494 100644
--- a/test/timestepping/heun.jl
+++ b/test/timestepping/heun.jl
@@ -146,3 +146,25 @@ end
     dt_heun = default_dt(integrator_heun)
     @test u_inner_heun[2] == (0.1 * dt_heun + (0.1 * dt_heun + 0.1) * dt_heun) / 2
 end
+
+# The cached fields must live on the state's grid object rather than on a copy of it: field operations
+# that combine state and cached fields compare grids, which is only cheap (and, under Reactant,
+# only traceable) when both refer to the identical grid.
+@testset "Heun cache shares the grid of the state" begin
+    grid = ColumnGrid(CPU(), Float64, UniformSpacing(Δz = 0.1, N = 1))
+    model = NamespacedExpModel(grid; timestepper = Heun())
+    integrator = initialize(model; initializers = (u = 0.0, v = 0.1))
+    state = integrator.state
+    cache = state.timestepper_cache
+    @test cache isa Terrarium.HeunCache
+
+    for (cached, current) in (
+            (cache.prognostic.u, state.prognostic.u),
+            (cache.tendencies.u, state.tendencies.u),
+            (cache.namespaces.inner.prognostic.u, state.namespaces.inner.prognostic.u),
+            (cache.namespaces.inner.tendencies.u, state.namespaces.inner.tendencies.u),
+        )
+        @test cached.grid === current.grid
+        @test parent(cached) !== parent(current)
+    end
+end