diff --git a/src/algorithms/contractions/ctmrg/characteristic_equations.jl b/src/algorithms/contractions/ctmrg/characteristic_equations.jl
index 621f37316..724e639d0 100644
--- a/src/algorithms/contractions/ctmrg/characteristic_equations.jl
+++ b/src/algorithms/contractions/ctmrg/characteristic_equations.jl
@@ -53,6 +53,7 @@ end
         Efp::EdgeTensor, # unused
         Ufp::LeftProjector,
         ULfp::LeftProjector;
+        projector_alg,
         getsite = getindex,
     )
 
@@ -66,7 +67,9 @@ corner and edge tensors, while ``u`` parametrizes a differentiable projector ``U
 U_{fp} + U_{L,fp} * u``.
 
 The local sandwich is read through `getsite`, so a caller can choose whether it is
-differentiated at all; see [`PEPSKit.constant_site`](@ref).
+differentiated at all; see [`PEPSKit.constant_site`](@ref). The projector algorithm
+`projector_alg` of the contraction selects how ``C_{fp}^{-1}`` is computed: the corner is
+diagonal for [`C4vEighProjector`](@ref), but only Hermitian for [`C4vQRProjector`](@ref).
 
 ``F`` returns a tuple of three tensors, corresponding to an equation for ``C``, ``E`` and
 ``u`` respectively:
@@ -100,10 +103,11 @@ function generate_symmetric_characteristic_equation(
         Efp::EdgeTensor, # unused
         Ufp::LeftProjector,
         ULfp::LeftProjector;
+        projector_alg,
         getsite = getindex,
     )
 
-    iC = sdiag_pow(real(DiagonalTensorMap(Cfp)), -1)
+    iC = _c4v_corner_inverse(projector_alg, Cfp)
     ULd = ULfp'
 
     function symmetric_characteristic_equation(n, C, E, u)
diff --git a/src/algorithms/ctmrg/projectors/c4v_eigh.jl b/src/algorithms/ctmrg/projectors/c4v_eigh.jl
index 9e4903900..c990bfed7 100644
--- a/src/algorithms/ctmrg/projectors/c4v_eigh.jl
+++ b/src/algorithms/ctmrg/projectors/c4v_eigh.jl
@@ -35,6 +35,9 @@ function C4vEighProjector(; kwargs...)
 end
 PROJECTOR_SYMBOLS[:C4vEighProjector] = C4vEighProjector
 
+# diagonal inverse for diagonal corner
+_c4v_corner_inverse(::C4vEighProjector, C) = sdiag_pow(real(DiagonalTensorMap(C)), -1)
+
 """
 Compute the normalized and Hermitian-symmetrized C₄ᵥ enlarged corner.
 ```
diff --git a/src/algorithms/ctmrg/projectors/c4v_qr.jl b/src/algorithms/ctmrg/projectors/c4v_qr.jl
index 51dfbd4f8..987adb4aa 100644
--- a/src/algorithms/ctmrg/projectors/c4v_qr.jl
+++ b/src/algorithms/ctmrg/projectors/c4v_qr.jl
@@ -26,6 +26,12 @@ PROJECTOR_SYMBOLS[:C4vQRProjector] = C4vQRProjector
 
 decomposition_algorithm(alg::C4vQRProjector) = alg.decomposition_alg
 
+# full inverse for non-diagonal Hermitian corner
+function _c4v_corner_inverse(::C4vQRProjector, C)
+    D, V = eigh_full(C)
+    return V * sdiag_pow(D, -1) * V'
+end
+
 # no truncation
 _set_truncation(alg::C4vQRProjector, ::TruncationStrategy) = alg
 _set_decomposition_truncation(alg::C4vQRProjector, ::TruncationStrategy) = alg
diff --git a/src/algorithms/optimization/implicit_differentiation.jl b/src/algorithms/optimization/implicit_differentiation.jl
index d3a0fbfe3..f64b1b17e 100644
--- a/src/algorithms/optimization/implicit_differentiation.jl
+++ b/src/algorithms/optimization/implicit_differentiation.jl
@@ -504,9 +504,11 @@ function _rrule(
     # Two tapes: the linear solve only uses the environment cotangents, so reading the
     # sandwich through `constant_site` leaves the ket and bra cotangents unforced. The state
     # pullback runs once, after the solve, and gets its own tape.
-    F_tracked = generate_symmetric_characteristic_equation(C, E, U, UL)
+    F_tracked = generate_symmetric_characteristic_equation(
+        C, E, U, UL; projector_alg = alg.projector_alg
+    )
     F_untracked = generate_symmetric_characteristic_equation(
-        C, E, U, UL; getsite = constant_site
+        C, E, U, UL; projector_alg = alg.projector_alg, getsite = constant_site
     )
     network = InfiniteSquareNetwork(state)