diff --git a/src/engine/engine_core_constraint.c b/src/engine/engine_core_constraint.c
index 34b2892e..d0d14caa 100644
--- a/src/engine/engine_core_constraint.c
+++ b/src/engine/engine_core_constraint.c
@@ -455,6 +455,45 @@ void mj_mulJacTVec(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec)
 
 
 
+// multiply Jacobian transpose by vector, for one island
+void mj_mulJacTVec_island(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec, int island) {
+  // no island, call regular function
+  if (island < 0) {
+    mj_mulJacTVec(m, d, res, vec);
+    return;
+  }
+
+  // sizes
+  int vecnnz = d->island_efcnum[island];
+  int resnnz = d->island_dofnum[island];
+
+  // indices
+  int* vecind = d->island_efcind + d->island_efcadr[island];
+  int* resind = d->island_dofind + d->island_dofadr[island];
+
+  // sparse Jacobian
+  if (mj_isSparse(m)) {
+    for (int i=0; i < resnnz; i++) {
+      int row = resind[i];
+      int JTnnz = d->efc_JT_rownnz[row];
+      int JTrowadr = d->efc_JT_rowadr[row];
+      int* JTind = d->efc_JT_colind + JTrowadr;
+      mjtNum* JT = d->efc_JT + JTrowadr;
+      res[i] = mju_dotSparse2(vec, JT, vecnnz, vecind, JTnnz, JTind);
+    }
+  }
+
+  // dense Jacobian
+  else {
+    int nefc = d->nefc;
+    for (int i=0; i < resnnz; i++) {
+      res[i] = mju_dotSparse(vec, d->efc_JT + nefc*resind[i], vecnnz, vecind);
+    }
+  }
+}
+
+
+
 //--------------------- instantiate constraints by type --------------------------------------------
 
 // equality constraints
@@ -1772,6 +1811,10 @@ void mj_makeConstraint(const mjModel* m, mjData* d) {
     // supernodes of JT
     mju_superSparse(m->nv, d->efc_JT_rowsuper,
                     d->efc_JT_rownnz, d->efc_JT_rowadr, d->efc_JT_colind);
+  } else {
+    if (mjENABLED(mjENBL_ISLAND)) {
+      mju_transpose(d->efc_JT, d->efc_J, d->nefc, m->nv);
+    }
   }
 
   // compute diagApprox
diff --git a/src/engine/engine_core_constraint.h b/src/engine/engine_core_constraint.h
index afa3447c..3104a339 100644
--- a/src/engine/engine_core_constraint.h
+++ b/src/engine/engine_core_constraint.h
@@ -42,6 +42,10 @@ MJAPI void mj_mulJacVec(const mjModel* m, mjData* d, mjtNum* res, const mjtNum*
 MJAPI void mj_mulJacVec_island(const mjModel* m, mjData* d,
                                mjtNum* res, const mjtNum* vec, int island);
 
+// multiply Jacobian transposed by vector, for one island
+MJAPI void mj_mulJacTVec_island(const mjModel* m, mjData* d,
+                                mjtNum* res, const mjtNum* vec, int island);
+
 // multiply JacobianT by vector
 MJAPI void mj_mulJacTVec(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec);
 
diff --git a/test/engine/engine_core_constraint_test.cc b/test/engine/engine_core_constraint_test.cc
index af2df04c..a0c2a477 100644
--- a/test/engine/engine_core_constraint_test.cc
+++ b/test/engine/engine_core_constraint_test.cc
@@ -261,5 +261,67 @@ TEST_F(CoreConstraintTest, MulJacVecIsland) {
   mj_deleteModel(model);
 }
 
+TEST_F(CoreConstraintTest, MulJacTVecIsland) {
+  const std::string xml_path = GetTestDataFilePath(kIlslandEfcPath);
+  mjModel* model = mj_loadXML(xml_path.c_str(), nullptr, nullptr, 0);
+  mjData* data = mj_makeData(model);
+
+  // allocate vec_nv
+  mjtNum* vec_nv = (mjtNum*) mju_malloc(sizeof(mjtNum)*model->nv);
+
+  // iterate through dense and sparse
+  for (mjtJacobian sparsity : {mjJAC_DENSE, mjJAC_SPARSE}) {
+    model->opt.jacobian = sparsity;
+
+    // simulate for 0.3 seconds
+    mj_resetData(model, data);
+    while (data->time < 0.3) {
+      mj_step(model, data);
+    }
+    mj_forward(model, data);
+
+    // allocate vec_nefc, fill with arbitrary values
+    mjtNum* vec_nefc = (mjtNum*) mju_malloc(sizeof(mjtNum)*data->nefc);
+    for (int i=0; i < data->nefc; i++) {
+      vec_nefc[i] = 0.2 + 0.3*i;
+    }
+
+    // multiply by Jacobian: vec_nv = J^T * vec_nefc
+    mj_mulJacTVec(model, data, vec_nv, vec_nefc);
+
+    // iterate over islands
+    for (int i=0; i < data->nisland; i++) {
+      // allocate dof and efc vectors for island
+      int dofnum = data->island_dofnum[i];
+      mjtNum* vec_nvi = (mjtNum*)mju_malloc(sizeof(mjtNum) * dofnum);
+      int efcnum = data->island_efcnum[i];
+      mjtNum* vec_nefci = (mjtNum*)mju_malloc(sizeof(mjtNum) * efcnum);
+
+      // copy values into vec_nefci
+      int* efcind = data->island_efcind + data->island_efcadr[i];
+      for (int j=0; j < efcnum; j++) {
+        vec_nefci[j] = vec_nefc[efcind[j]];
+      }
+
+      // multiply by Jacobian, for this island
+      mj_mulJacTVec_island(model, data, vec_nvi, vec_nefci, i);
+
+      // expect corresponding values to match
+      int* dofind = data->island_dofind + data->island_dofadr[i];
+      for (int j=0; j < dofnum; j++) {
+        EXPECT_THAT(vec_nvi[j], DoubleNear(vec_nv[dofind[j]], 1e-12));
+      }
+
+      mju_free(vec_nvi);
+      mju_free(vec_nefci);
+    }
+    mju_free(vec_nefc);
+  }
+
+  mju_free(vec_nv);
+  mj_deleteData(data);
+  mj_deleteModel(model);
+}
+
 }  // namespace
 }  // namespace mujoco