diff --git a/libvips/foreign/tiff2vips.c b/libvips/foreign/tiff2vips.c
index 6e39a5f7..35046d81 100644
--- a/libvips/foreign/tiff2vips.c
+++ b/libvips/foreign/tiff2vips.c
@@ -1904,9 +1904,11 @@ vips__istiff_buffer( void *buf, size_t len )
 {
 	char *str = (char *) buf; 
 
-	if( len >= 2 &&
-		((str[0] == 'M' && str[1] == 'M') ||
-		 (str[0] == 'I' && str[1] == 'I')) ) 
+	if( len >= 4 &&
+		((str[0] == 'M' && str[1] == 'M' &&
+			str[2] == '\0' && (str[3] == '*' || str[3] == '+')) ||
+		(str[0] == 'I' && str[1] == 'I' &&
+			(str[2] == '*' || str[2] == '+') && str[3] == '\0')) )
 		return( TRUE );
 
 	return( FALSE );
@@ -1915,10 +1917,10 @@ vips__istiff_buffer( void *buf, size_t len )
 gboolean
 vips__istiff( const char *filename )
 {
-	unsigned char buf[2];
+	unsigned char buf[4];
 
-	if( vips__get_bytes( filename, buf, 2 ) &&
-		vips__istiff_buffer( buf, 2 ) )
+	if( vips__get_bytes( filename, buf, 4 ) &&
+		vips__istiff_buffer( buf, 4 ) )
 		return( TRUE );
 
 	return( FALSE );