diff --git a/swh/loader/git/loader.py b/swh/loader/git/loader.py
--- a/swh/loader/git/loader.py
+++ b/swh/loader/git/loader.py
@@ -97,21 +97,42 @@
             for name, value in refs.items():
                 heads_logger.debug("    %r: %s", name, value.decode())
 
-        # Get the remote heads that we want to fetch
-        remote_heads: Set[HexBytes] = set()
+        # specific set of objects to sort
+        tag_names = set()
+        branch_names = set()
+        # remote heads is just all refs without order
+        remote_heads = set()
+        target_to_ref = defaultdict(list)
         for ref_name, ref_target in refs.items():
-            if utils.ignore_branch_name(ref_name):
+            # Ignore either usual branch to ignore or known references
+            if utils.ignore_branch_name(ref_name) or ref_target in self.local_heads:
                 continue
-            remote_heads.add(ref_target)
+            # Then we'll sort out the tags from the branches
+            if ref_name.startswith(b"refs/tags/"):
+                tag_names.add(ref_name)
+            else:
+                branch_names.add(ref_name)
 
-        if heads_logger.isEnabledFor(logging.DEBUG):
-            heads_logger.debug("Filtered remote heads:")
-            for value in remote_heads:
-                heads_logger.debug("    %s", value.decode())
+            remote_heads.add(ref_target)
+            target_to_ref[ref_target].append(ref_name)
 
         logger.debug("local_heads_count=%s", len(self.local_heads))
         logger.debug("remote_heads_count=%s", len(remote_heads))
-        wanted_refs = list(remote_heads - self.local_heads)
+
+        # Then we sort the refs (by tags then by branches) so it's mostly ingested in
+        # lexicographic order (provided there is some consistency there)
+        tags = [refs[ref_name] for ref_name in sorted(tag_names)]
+        branches = [refs[ref_name] for ref_name in sorted(branch_names)]
+        # The wanted refs is the concatenation first tags then branches references
+        wanted_refs = tags + branches
+
+        if heads_logger.isEnabledFor(logging.DEBUG):
+            heads_logger.debug("Ordered wanted heads returned by the git remote:")
+            for ref_target in wanted_refs:
+                heads_logger.debug(
+                    "    %r: %s", target_to_ref[ref_target], ref_target.decode()
+                )
+
         logger.debug("wanted_refs_count=%s", len(wanted_refs))
         if self.statsd is not None:
             self.statsd.histogram(
@@ -119,9 +140,14 @@
                 len(remote_heads - set(refs.values())) / len(refs),
                 tags={},
             )
+            git_known_refs_percent = (
+                len(self.local_heads & remote_heads) / len(remote_heads)
+                if remote_heads
+                else 0
+            )
             self.statsd.histogram(
                 "git_known_refs_percent",
-                len(self.local_heads & remote_heads) / len(remote_heads),
+                git_known_refs_percent,
                 tags={},
             )
         return wanted_refs
diff --git a/swh/loader/git/tests/test_loader.py b/swh/loader/git/tests/test_loader.py
--- a/swh/loader/git/tests/test_loader.py
+++ b/swh/loader/git/tests/test_loader.py
@@ -452,7 +452,7 @@
         assert [c for c in statsd_report.mock_calls if c[1][0].startswith("git_")] == [
             call("git_total", "c", 1, {}, 1),
             call("git_ignored_refs_percent", "h", 0.0, {}, 1),
-            call("git_known_refs_percent", "h", 0.25, {}, 1),
+            call("git_known_refs_percent", "h", 0.0, {}, 1),
         ]
         assert self.loader.statsd.constant_tags == {
             "visit_type": "git",
@@ -512,7 +512,7 @@
         assert [c for c in statsd_report.mock_calls if c[1][0].startswith("git_")] == [
             call("git_total", "c", 1, {}, 1),
             call("git_ignored_refs_percent", "h", 0.0, {}, 1),
-            call("git_known_refs_percent", "h", 1.0, {}, 1),
+            call("git_known_refs_percent", "h", 0.0, {}, 1),
         ]
         assert self.loader.statsd.constant_tags == {
             "visit_type": "git",
@@ -532,7 +532,7 @@
                     }
                 ),
                 Snapshot(branches={}),
-                0.25,
+                0.0,
                 id="partial-parent-and-empty-previous",
             ),
             pytest.param(
@@ -542,7 +542,7 @@
                         b"refs/heads/master": SNAPSHOT1.branches[b"refs/heads/master"]
                     }
                 ),
-                1.0,
+                0.0,
                 id="full-parent-and-partial-previous",
             ),
         ],