diff --git a/src/solver/impls/snes/snes.cxx b/src/solver/impls/snes/snes.cxx
index 446903dd4f..2fc8b8ef5a 100644
--- a/src/solver/impls/snes/snes.cxx
+++ b/src/solver/impls/snes/snes.cxx
@@ -604,7 +604,9 @@ SNESSolver::SNESSolver(Options* opts)
                      .withDefault<bool>(false)),
       asinh_vars((*options)["asinh_vars"]
                      .doc("Apply asinh() to all variables?")
-                     .withDefault<bool>(false)) {}
+                     .withDefault<bool>(false)) {
+  has_constraints = true; ///< This solver can handle constraints
+}
 
 int SNESSolver::init() {
 
@@ -627,6 +629,29 @@ int SNESSolver::init() {
   output_info.write("\t3d fields = {:d}, 2d fields = {:d} neq={:d}, local_N={:d}\n",
                     n3Dvars(), n2Dvars(), neq, nlocal);
 
+  // Check if there are any constraints
+  have_constraints = false;
+
+  for (int i = 0; i < n2Dvars(); i++) {
+    if (f2d[i].constraint) {
+      have_constraints = true;
+      break;
+    }
+  }
+  for (int i = 0; i < n3Dvars(); i++) {
+    if (f3d[i].constraint) {
+      have_constraints = true;
+      break;
+    }
+  }
+
+  if (have_constraints) {
+    is_dae.reallocate(nlocal);
+    // Call the Solver function, which sets the array
+    // to one when not a constraint, zero for constraint
+    set_id(std::begin(is_dae));
+  }
+
   // Initialise PETSc components
 
   // Vectors
@@ -673,6 +698,34 @@ int SNESSolver::init() {
     PetscCall(initPseudoTimestepping());
   }
 
+  if (have_constraints) {
+    // CreatePETSc-native index sets representing the two parts of your DAE.
+    PetscInt istart, iend;
+    PetscCall(VecGetOwnershipRange(snes_x, &istart, &iend));
+    ASSERT2(iend - istart == nlocal);
+
+    std::vector<PetscInt> diff_idx;
+    std::vector<PetscInt> alg_idx;
+    diff_idx.reserve(nlocal);
+    alg_idx.reserve(nlocal);
+
+    for (PetscInt i = 0; i < nlocal; ++i) {
+      const PetscInt gi = istart + i;
+      if (is_dae[i] > 0.5) { // differential
+        diff_idx.push_back(gi);
+      } else { // algebraic constraint (i.e. phi)
+        alg_idx.push_back(gi);
+      }
+    }
+
+    PetscCall(ISCreateGeneral(BoutComm::get(), diff_idx.size(), diff_idx.data(),
+                              PETSC_COPY_VALUES, &is_diff));
+    PetscCall(ISCreateGeneral(BoutComm::get(), alg_idx.size(), alg_idx.data(),
+                              PETSC_COPY_VALUES, &is_alg));
+
+    have_is_maps = true;
+  }
+
   // Nonlinear solver interface (SNES)
   output_info.write("Create SNES\n");
   SNESCreate(BoutComm::get(), &snes);
@@ -753,6 +806,9 @@ int SNESSolver::init() {
   SNESSetForceIteration(snes, PETSC_TRUE);
 #endif
 
+  // Enable checking for domain errors in Jacobian evaluation
+  SNESSetCheckJacobianDomainError(snes, PETSC_TRUE);
+
   // Get KSP context from SNES
   KSP ksp;
   SNESGetKSP(snes, &ksp);
@@ -805,6 +861,24 @@ int SNESSolver::init() {
     }
   }
 
+  if (have_constraints && have_is_maps && !matrix_free && pc_type == "fieldsplit") {
+    output_info.write("Using PCFieldSplit preconditioner for DAE system\n");
+
+    // Use PETSc fieldsplit
+    PetscCall(PCSetType(pc, PCFIELDSPLIT));
+
+    // Give PETSc the index sets
+    PetscCall(PCFieldSplitSetIS(pc, "diff", is_diff));
+    PetscCall(PCFieldSplitSetIS(pc, "alg", is_alg));
+
+    // Let the user configure from options (recommended)
+    // Example options you can set in input file:
+    //   -pc_fieldsplit_type additive
+    //   -fieldsplit_alg_pc_type hypre -fieldsplit_alg_pc_hypre_type boomeramg
+    //   -fieldsplit_diff_pc_type ilu
+    //
+  }
+
   // Get runtime options
   lib.setOptionsFromInputFile(snes);
 
@@ -1162,7 +1236,7 @@ int SNESSolver::run() {
 
         timestep = pid(timestep, nl_its, max_timestep);
 
-        // NOTE(malamast): Do we really need this?
+        // NOTE: Do we really need this?
         // Recompute Jacobian (for now)
         if (saved_jacobian_lag == 0) {
           SNESGetLagJacobian(snes, &saved_jacobian_lag);
@@ -1262,6 +1336,8 @@ int SNESSolver::run() {
       run_rhs(target); // Run RHS to calculate auxilliary variables
     } catch (BoutException& e) {
       output_error.write("ERROR: BoutException thrown: {}\n", e.what());
+      // NOTE: what happens if we hit the exception here?
+      // Should we add a relaxation step to update the state vector?
     }
 
     if (call_monitors(target, s, getNumberOutputSteps()) != 0) {
@@ -1573,7 +1649,13 @@ PetscErrorCode SNESSolver::snes_function(Vec x, Vec f, bool linear) {
   // Call the RHS function
   if (rhs_function(x, f, linear) != PETSC_SUCCESS) {
     // Tell SNES that the input was out of domain
-    SNESSetFunctionDomainError(snes);
+    if (linear) {
+      // During Jacobian evaluation
+      SNESSetJacobianDomainError(snes);
+    } else {
+      // During function evaluation
+      SNESSetFunctionDomainError(snes);
+    }
     // Note: Returning non-zero error here leaves vectors in locked state
     return 0;
   }
@@ -1597,10 +1679,33 @@ PetscErrorCode SNESSolver::snes_function(Vec x, Vec f, bool linear) {
     break;
   }
   case BoutSnesEquationForm::backward_euler: {
-    // Backward Euler
-    // Set f = x - x0 - Δt*f
-    VecAYPX(f, -dt, x);   // f <- x - Δt*f
-    VecAXPY(f, -1.0, x0); // f <- f - x0
+    // Backward Euler:
+    // Differential:  F = x - x0 - dt*f
+    // Algebraic:     F = G(x)  (already stored in f by rhs_function)
+
+    if (!have_constraints) {
+
+      VecAYPX(f, -dt, x);   // f <- x - Δt*f
+      VecAXPY(f, -1.0, x0); // f <- f - x0
+
+    } else {
+
+      ASSERT2(have_is_maps);
+      // Some constraints
+
+      Vec x_diff, x0_diff, f_diff;
+
+      PetscCall(VecGetSubVector(x, is_diff, &x_diff));
+      PetscCall(VecGetSubVector(x0, is_diff, &x0_diff));
+      PetscCall(VecGetSubVector(f, is_diff, &f_diff));
+
+      PetscCall(VecAYPX(f_diff, -dt, x_diff));   // f_diff <- x_diff - dt*f_diff
+      PetscCall(VecAXPY(f_diff, -1.0, x0_diff)); // f_diff <- f_diff - x0_diff
+
+      PetscCall(VecRestoreSubVector(x, is_diff, &x_diff));
+      PetscCall(VecRestoreSubVector(x0, is_diff, &x0_diff));
+      PetscCall(VecRestoreSubVector(f, is_diff, &f_diff));
+    }
     break;
   }
   case BoutSnesEquationForm::direct_newton: {
diff --git a/src/solver/impls/snes/snes.hxx b/src/solver/impls/snes/snes.hxx
index 5d007f8ad7..1ea06f6481 100644
--- a/src/solver/impls/snes/snes.hxx
+++ b/src/solver/impls/snes/snes.hxx
@@ -182,6 +182,13 @@ private:
   int nlocal; ///< Number of variables on local processor
   int neq;    ///< Number of variables in total
 
+  bool have_constraints;  ///< Are there any constraint variables?
+  Array<BoutReal> is_dae; ///< If using constraints, 1 -> DAE, 0 -> AE
+
+  IS is_diff = nullptr; // is_dae == 1
+  IS is_alg = nullptr;  // is_dae == 0 (phi constraint and any other algebraics)
+  bool have_is_maps = false;
+
   PetscLib lib; ///< Handles initialising, finalising PETSc
   Vec snes_f;   ///< Used by SNES to store function
   Vec snes_x;   ///< Result of SNES