diff --git a/src/grids.cpp b/src/grids.cpp
index 624f2dfa13..09e9b9fe3d 100644
--- a/src/grids.cpp
+++ b/src/grids.cpp
@@ -1320,6 +1320,15 @@ std::unique_ptr<GTiffGrid> GTiffDataset::nextGrid() {
     m_ifdIdx++;
     m_hasNextGrid = TIFFReadDirectory(m_hTIFF) != 0;
     m_nextDirOffset = TIFFCurrentDirOffset(m_hTIFF);
+
+    // If the TIFF file contains multiple grids, append the index of the grid
+    // in the grid name to help debugging.
+    if (m_ifdIdx >= 2 || m_hasNextGrid) {
+        ret->m_name += " (index ";
+        ret->m_name += std::to_string(m_ifdIdx); // 1-based
+        ret->m_name += ')';
+    }
+
     return ret;
 }
 
diff --git a/src/grids.hpp b/src/grids.hpp
index a5e0d8c4c4..a7b5946976 100644
--- a/src/grids.hpp
+++ b/src/grids.hpp
@@ -59,6 +59,7 @@ struct ExtentAndRes {
 
 class PROJ_GCC_DLL Grid {
   protected:
+    friend class GTiffDataset;
     std::string m_name;
     int m_width;
     int m_height;