tpl: Fix issue with partials without suffix

Fixes #13601

tpl/templates/templates_integration_test.go

@@ -279,3 +279,23 @@ P1.
 	b.Assert(err, qt.IsNotNil)
 	b.Assert(err.Error(), qt.Contains, "wrong number of args for string: want 1 got 0")
 }
+
+func TestPartialWithoutSuffixIssue13601(t *testing.T) {
+	t.Parallel()
+
+	files := `
+-- hugo.toml --
+-- layouts/home.html --
+P1: {{ partial "p1" . }}
+P2: {{ partial "p2" . }}
+-- layouts/_partials/p1 --
+P1.
+-- layouts/_partials/p2 --
+P2.
+{{ return "foo bar" }}
+
+`
+
+	b := hugolib.Test(t, files)
+	b.AssertFileContent("public/index.html", "P1: P1.\nP2: foo bar")
+}

tpl/tplimpl/templatestore.go

@@ -913,7 +913,7 @@ func (s *TemplateStore) extractInlinePartials() error {
 			name := templ.Name()
 			if !paths.HasExt(name) {
 				// Assume HTML. This in line with how the lookup works.
-				name = name + ".html"
+				name = name + s.htmlFormat.MediaType.FirstSuffix.FullSuffix
 			}
 			if !strings.HasPrefix(name, "_") {
 				name = "_" + name
@@ -1092,6 +1092,12 @@ func (s *TemplateStore) insertTemplate2(
 		panic("category not set")
 	}
 
+	if category == CategoryPartial && d.OutputFormat == "" && d.MediaType == "" {
+		// See issue #13601.
+		d.OutputFormat = s.htmlFormat.Name
+		d.MediaType = s.htmlFormat.MediaType.Type
+	}
+
 	m := tree.Get(key)
 	nk := nodeKey{c: category, d: d}
 
@@ -1719,6 +1725,7 @@ func (s *TemplateStore) transformTemplates() error {
 			continue
 		}
 		if !vv.noBaseOf {
+			// TODO(bep) I don't think this branch is ever called.
 			for vvv := range vv.BaseVariantsSeq() {
 				tctx, err := applyTemplateTransformers(vvv.Template, lookup)
 				if err != nil {