diff --git a/src/Magnum/GL/Implementation/MeshState.cpp b/src/Magnum/GL/Implementation/MeshState.cpp
index 6fecd4388..f9719984c 100644
--- a/src/Magnum/GL/Implementation/MeshState.cpp
+++ b/src/Magnum/GL/Implementation/MeshState.cpp
@@ -277,12 +277,18 @@ MeshState::MeshState(Context& context, ContextState& contextState, Containers::S
            1 at all. */
         #if !(defined(MAGNUM_TARGET_WEBGL) && defined(MAGNUM_TARGET_GLES2))
         #ifndef MAGNUM_TARGET_WEBGL
-        if(context.isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()) {
+        /* The {EXT,OES}_draw_elements_base_vertex supports the multidraw
+           entrypoint only if EXT_multi_draw_arrays is supported as well. Which
+           is the case on Mesa, for example, but not on ANGLE -- there we have
+           to use the entrypoint from ANGLE_base_vertex_base_instance instead. */
+        if(context.isExtensionSupported<Extensions::EXT::multi_draw_arrays>() &&
+           context.isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()) {
             extensions[Extensions::EXT::draw_elements_base_vertex::Index] =
                        Extensions::EXT::draw_elements_base_vertex::string();
 
             multiDrawElementsBaseVertexImplementation = glMultiDrawElementsBaseVertexEXT;
-        } else if(context.isExtensionSupported<Extensions::OES::draw_elements_base_vertex>()) {
+        } else if(context.isExtensionSupported<Extensions::EXT::multi_draw_arrays>() &&
+           context.isExtensionSupported<Extensions::OES::draw_elements_base_vertex>()) {
             extensions[Extensions::OES::draw_elements_base_vertex::Index] =
                        Extensions::OES::draw_elements_base_vertex::string();
 
diff --git a/src/Magnum/GL/Test/MeshGLTest.cpp b/src/Magnum/GL/Test/MeshGLTest.cpp
index 74a368104..6aad7fb98 100644
--- a/src/Magnum/GL/Test/MeshGLTest.cpp
+++ b/src/Magnum/GL/Test/MeshGLTest.cpp
@@ -4051,15 +4051,22 @@ template<class T> void MeshGLTest::multiDrawIndexed() {
             CORRADE_SKIP(Extensions::ARB::draw_elements_base_vertex::string() << "is not supported.");
         #elif !defined(MAGNUM_TARGET_WEBGL)
         #ifndef MAGNUM_TARGET_GLES2
-        if(!Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::ANGLE::base_vertex_base_instance>()
+        /* {OES,EXT}_draw_elements_base_vertex requires EXT_multi_draw_arrays
+           for the multi-draw entrypoint */
+        if((!Context::current().isExtensionSupported<Extensions::EXT::multi_draw_arrays>() || (
+            !Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
+            !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
+        )) &&
+            !Context::current().isExtensionSupported<Extensions::ANGLE::base_vertex_base_instance>()
         )
             CORRADE_SKIP("Neither" << Extensions::OES::draw_elements_base_vertex::string() << "nor" << Extensions::EXT::draw_elements_base_vertex::string() << "nor" << Extensions::ANGLE::base_vertex_base_instance::string() << "is supported.");
         #else
-        if(!Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
-        )
+        /* {OES,EXT}_draw_elements_base_vertex requires EXT_multi_draw_arrays
+           for the multi-draw entrypoint */
+        if(!Context::current().isExtensionSupported<Extensions::EXT::multi_draw_arrays>() || (
+            !Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
+            !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
+        ))
             CORRADE_SKIP("Neither" << Extensions::OES::draw_elements_base_vertex::string() << "nor" << Extensions::EXT::draw_elements_base_vertex::string() << "is supported.");
         #endif
         #elif !defined(MAGNUM_TARGET_GLES2)
@@ -4138,15 +4145,22 @@ template<class T> void MeshGLTest::multiDrawIndexedSparseArrays() {
             CORRADE_SKIP(Extensions::ARB::draw_elements_base_vertex::string() << "is not supported.");
         #elif !defined(MAGNUM_TARGET_WEBGL)
         #ifndef MAGNUM_TARGET_GLES2
-        if(!Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::ANGLE::base_vertex_base_instance>()
+        /* {OES,EXT}_draw_elements_base_vertex requires EXT_multi_draw_arrays
+           for the multi-draw entrypoint */
+        if((!Context::current().isExtensionSupported<Extensions::EXT::multi_draw_arrays>() || (
+            !Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
+            !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
+        )) &&
+            !Context::current().isExtensionSupported<Extensions::ANGLE::base_vertex_base_instance>()
         )
             CORRADE_SKIP("Neither" << Extensions::OES::draw_elements_base_vertex::string() << "nor" << Extensions::EXT::draw_elements_base_vertex::string() << "nor" << Extensions::ANGLE::base_vertex_base_instance::string() << "is supported.");
         #else
-        if(!Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
-        )
+        /* {OES,EXT}_draw_elements_base_vertex requires EXT_multi_draw_arrays
+           for the multi-draw entrypoint */
+        if(!Context::current().isExtensionSupported<Extensions::EXT::multi_draw_arrays>() || (
+            !Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
+            !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
+        ))
             CORRADE_SKIP("Neither" << Extensions::OES::draw_elements_base_vertex::string() << "nor" << Extensions::EXT::draw_elements_base_vertex::string() << "is supported.");
         #endif
         #elif !defined(MAGNUM_TARGET_GLES2)
@@ -4221,15 +4235,22 @@ void MeshGLTest::multiDrawIndexedViews() {
             CORRADE_SKIP(Extensions::ARB::draw_elements_base_vertex::string() << "is not supported.");
         #elif !defined(MAGNUM_TARGET_WEBGL)
         #ifndef MAGNUM_TARGET_GLES2
-        if(!Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::ANGLE::base_vertex_base_instance>()
+        /* {OES,EXT}_draw_elements_base_vertex requires EXT_multi_draw_arrays
+           for the multi-draw entrypoint */
+        if((!Context::current().isExtensionSupported<Extensions::EXT::multi_draw_arrays>() || (
+            !Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
+            !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
+        )) &&
+            !Context::current().isExtensionSupported<Extensions::ANGLE::base_vertex_base_instance>()
         )
             CORRADE_SKIP("Neither" << Extensions::OES::draw_elements_base_vertex::string() << "nor" << Extensions::EXT::draw_elements_base_vertex::string() << "nor" << Extensions::ANGLE::base_vertex_base_instance::string() << "is supported.");
         #else
-        if(!Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
-           !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
-        )
+        /* {OES,EXT}_draw_elements_base_vertex requires EXT_multi_draw_arrays
+           for the multi-draw entrypoint */
+        if(!Context::current().isExtensionSupported<Extensions::EXT::multi_draw_arrays>() || (
+            !Context::current().isExtensionSupported<Extensions::OES::draw_elements_base_vertex>() &&
+            !Context::current().isExtensionSupported<Extensions::EXT::draw_elements_base_vertex>()
+        ))
             CORRADE_SKIP("Neither" << Extensions::OES::draw_elements_base_vertex::string() << "nor" << Extensions::EXT::draw_elements_base_vertex::string() << "is supported.");
         #endif
         #elif !defined(MAGNUM_TARGET_GLES2)