diff --git a/src/transformations/gridshift.cpp b/src/transformations/gridshift.cpp
index 260b5b05ca..ab1470b842 100644
--- a/src/transformations/gridshift.cpp
+++ b/src/transformations/gridshift.cpp
@@ -909,37 +909,41 @@ PJ *PJ_TRANSFORMATION(gridshift, 0) {
         return pj_gridshift_destructor(P, PROJ_ERR_INVALID_OP_MISSING_ARG);
     }
 
+    bool isKnownGrid = false;
     bool isProjectedCoord = false;
-    if (P->ctx->defer_grid_opening) {
-        Q->m_defer_grid_opening = true;
-    } else {
+
+    if (!P->ctx->defer_grid_opening ||
+        !pj_param(P->ctx, P->params, "tcoord_type").i) {
         const char *gridnames = pj_param(P->ctx, P->params, "sgrids").s;
         gMutex.lock();
         const auto iter = gKnownGrids.find(gridnames);
-        const bool isKnownGrid = iter != gKnownGrids.end();
+        isKnownGrid = iter != gKnownGrids.end();
         if (isKnownGrid) {
+            Q->m_defer_grid_opening = true;
             isProjectedCoord = iter->second;
         }
         gMutex.unlock();
-        if (isKnownGrid) {
-            Q->m_defer_grid_opening = true;
-        } else {
-            Q->m_grids = pj_generic_grid_init(P, "grids");
-            /* Was gridlist compiled properly? */
-            if (proj_errno(P)) {
-                proj_log_error(P, _("could not find required grid(s)."));
-                return pj_gridshift_destructor(
-                    P, PROJ_ERR_INVALID_OP_FILE_NOT_FOUND_OR_INVALID);
-            }
-            if (!Q->checkGridTypes(P, isProjectedCoord)) {
-                return pj_gridshift_destructor(
-                    P, PROJ_ERR_INVALID_OP_FILE_NOT_FOUND_OR_INVALID);
-            }
+    }
 
-            gMutex.lock();
-            gKnownGrids[gridnames] = isProjectedCoord;
-            gMutex.unlock();
+    if (P->ctx->defer_grid_opening || isKnownGrid) {
+        Q->m_defer_grid_opening = true;
+    } else {
+        const char *gridnames = pj_param(P->ctx, P->params, "sgrids").s;
+        Q->m_grids = pj_generic_grid_init(P, "grids");
+        /* Was gridlist compiled properly? */
+        if (proj_errno(P)) {
+            proj_log_error(P, _("could not find required grid(s)."));
+            return pj_gridshift_destructor(
+                P, PROJ_ERR_INVALID_OP_FILE_NOT_FOUND_OR_INVALID);
+        }
+        if (!Q->checkGridTypes(P, isProjectedCoord)) {
+            return pj_gridshift_destructor(
+                P, PROJ_ERR_INVALID_OP_FILE_NOT_FOUND_OR_INVALID);
         }
+
+        gMutex.lock();
+        gKnownGrids[gridnames] = isProjectedCoord;
+        gMutex.unlock();
     }
 
     if (pj_param(P->ctx, P->params, "tinterpolation").i) {
@@ -994,12 +998,17 @@ PJ *PJ_TRANSFORMATION(gridshift, 0) {
         }
     }
 
-    if (isProjectedCoord) {
-        P->left = PJ_IO_UNITS_PROJECTED;
-        P->right = PJ_IO_UNITS_PROJECTED;
+    if (isKnownGrid || pj_param(P->ctx, P->params, "tcoord_type").i) {
+        if (isProjectedCoord) {
+            P->left = PJ_IO_UNITS_PROJECTED;
+            P->right = PJ_IO_UNITS_PROJECTED;
+        } else {
+            P->left = PJ_IO_UNITS_RADIANS;
+            P->right = PJ_IO_UNITS_RADIANS;
+        }
     } else {
-        P->left = PJ_IO_UNITS_RADIANS;
-        P->right = PJ_IO_UNITS_RADIANS;
+        P->left = PJ_IO_UNITS_WHATEVER;
+        P->right = PJ_IO_UNITS_WHATEVER;
     }
 
     return P;
diff --git a/test/CMakeLists.txt b/test/CMakeLists.txt
index 9ec53dc567..2b2141a32f 100644
--- a/test/CMakeLists.txt
+++ b/test/CMakeLists.txt
@@ -60,6 +60,8 @@ proj_add_gie_test("Deformation" "gie/deformation.gie")
 proj_add_gie_test("geotiff_grids" "gie/geotiff_grids.gie")
 proj_add_gie_test("defmodel" "gie/defmodel.gie")
 proj_add_gie_test("gridshift" "gie/gridshift.gie")
+# Does not really require the network, but test running with PROJ_NETWORK=ON
+proj_add_gie_network_dependent_test("gridshift_network_enabled" "gie/gridshift.gie")
 endif()
 
 if(TIFF_ENABLED AND CURL_ENABLED AND RUN_NETWORK_DEPENDENT_TESTS)
diff --git a/test/gie/gridshift.gie b/test/gie/gridshift.gie
index fec8e50106..98cf0657aa 100644
--- a/test/gie/gridshift.gie
+++ b/test/gie/gridshift.gie
@@ -251,6 +251,19 @@ accept      -598000.000  -1160020.000 0.000
 expect      -5597999.885 -6160019.978 0.000
 roundtrip   1
 
+-------------------------------------------------------------------------------
+operation   +proj=pipeline \
+                +step +proj=krovak +lat_0=49.5 +lon_0=24.8333333333333 +alpha=30.2881397527778 +k=0.9999 +x_0=0 +y_0=0 +ellps=bessel \
+                +step +proj=gridshift +grids=tests/test_gridshift_projected.tif \
+                +step +inv +proj=mod_krovak +lat_0=49.5 +lon_0=24.8333333333333 +alpha=30.2881397222222 +k=0.9999 +x_0=5000000 +y_0=5000000 +ellps=bessel
+-------------------------------------------------------------------------------
+
+tolerance   0.5 mm
+
+accept      16.610452439 49.202425040 0.000
+expect      16.610455233 49.202425034 0.000
+roundtrip   1
+
 -------------
 # Error cases
 -------------