diff --git a/swh/deposit/api/checks.py b/swh/deposit/api/checks.py
--- a/swh/deposit/api/checks.py
+++ b/swh/deposit/api/checks.py
@@ -32,6 +32,8 @@
 SUGGESTED_FIELDS_MISSING = "Suggested fields are missing"
 METADATA_PROVENANCE_KEY = "swh:metadata-provenance"
 
+AFFILIATION_NO_NAME = "Reason: affiliation does not have a <codemeta:name> element"
+
 
 def extra_validator(element, xsd_element):
     """Performs extra checks on Atom elements that cannot be implemented purely
@@ -142,6 +144,44 @@
             schemas().codemeta.validate(child, extra_validator=extra_validator)
         except xmlschema.exceptions.XMLSchemaException as e:
             detail.append({"fields": [schema_element.prefixed_name], "summary": str(e)})
+        else:
+            # Manually validate <codemeta:affiliation>. Unfortunately, this cannot be
+            # validated by codemeta.xsd, because Codemeta has conflicting requirements:
+            # 1. https://codemeta.github.io/terms/ requires it to be Text (represented
+            #    by simple content), but
+            # 2. https://doi.org/10.5063/SCHEMA/CODEMETA-2.0 requires it to be an
+            #    Organization (represented by complex content)
+            # And this is (legitimately) not representable in XML Schema.
+            #
+            # See https://github.com/codemeta/codemeta/pull/239 for a discussion about
+            # this issue.
+            for affiliation in child.findall(
+                "codemeta:affiliation", namespaces=NAMESPACES
+            ):
+                if len(affiliation) > 0:
+                    # This is a complex element (as required by
+                    # https://codemeta.github.io/terms/), then we want to make sure
+                    # there is at least a name.
+                    if not affiliation.findtext("codemeta:name", namespaces=NAMESPACES):
+                        detail.append(
+                            {
+                                "fields": [schema_element.prefixed_name],
+                                "summary": AFFILIATION_NO_NAME,
+                            }
+                        )
+                        break
+                else:
+                    # This is a simple element (as required by
+                    # https://doi.org/10.5063/SCHEMA/CODEMETA-2.0)
+                    if affiliation.text is None or not affiliation.text.strip():
+                        # Completely empty element
+                        detail.append(
+                            {
+                                "fields": [schema_element.prefixed_name],
+                                "summary": AFFILIATION_NO_NAME,
+                            }
+                        )
+                        break
 
     if detail:
         return False, {"metadata": detail + suggested_fields}
diff --git a/swh/deposit/tests/api/test_checks.py b/swh/deposit/tests/api/test_checks.py
--- a/swh/deposit/tests/api/test_checks.py
+++ b/swh/deposit/tests/api/test_checks.py
@@ -226,7 +226,9 @@
             """,
         ),
         (
-            "codemeta-affiliation",
+            # Required by codemeta.jsonld, but forbidden by
+            # https://codemeta.github.io/terms/
+            "element-in--affiliation",
             f"""\
             <entry {XMLNS}>
                 <url>some url</url>
@@ -241,6 +243,22 @@
             </entry>
             """,
         ),
+        (
+            # Forbidden by codemeta.jsonld, but required by
+            # https://codemeta.github.io/terms/
+            "chardata-in-affiliation",
+            f"""\
+            <entry {XMLNS}>
+                <url>some url</url>
+                <codemeta:name>bar</codemeta:name>
+                <codemeta:author>
+                    <codemeta:name>someone</codemeta:name>
+                    <codemeta:affiliation>My Orga</codemeta:affiliation>
+                </codemeta:author>
+                {PROVENANCE_XML}
+            </entry>
+            """,
+        ),
         (
             "swh:add_to_origin",
             f"""\
@@ -494,21 +512,44 @@
             ],
         ),
         (
-            "chardata-in-affiliation",
+            "affiliation-with-no-name",
             f"""\
             <entry {XMLNS}>
                 <url>some url</url>
                 <codemeta:name>bar</codemeta:name>
                 <codemeta:author>
                     <codemeta:name>someone</codemeta:name>
-                    <codemeta:affiliation>My Orga</codemeta:affiliation>
+                    <codemeta:affiliation>
+                        <codemeta:url>http://example.org</codemeta:url>
+                    </codemeta:affiliation>
                 </codemeta:author>
                 {PROVENANCE_XML}
             </entry>
             """,
             [
                 {
-                    "summary": ".*Reason: character data between child elements.*",
+                    "summary": ".*Reason: affiliation does not have a <codemeta:name> element.*",
+                    "fields": ["codemeta:author"],
+                },
+            ],
+        ),
+        (
+            "empty-affiliation",
+            f"""\
+            <entry {XMLNS}>
+                <url>some url</url>
+                <codemeta:name>bar</codemeta:name>
+                <codemeta:author>
+                    <codemeta:name>someone</codemeta:name>
+                    <codemeta:affiliation>
+                    </codemeta:affiliation>
+                </codemeta:author>
+                {PROVENANCE_XML}
+            </entry>
+            """,
+            [
+                {
+                    "summary": ".*Reason: affiliation does not have a <codemeta:name> element.*",
                     "fields": ["codemeta:author"],
                 },
             ],
diff --git a/swh/deposit/xsd/codemeta.xsd b/swh/deposit/xsd/codemeta.xsd
--- a/swh/deposit/xsd/codemeta.xsd
+++ b/swh/deposit/xsd/codemeta.xsd
@@ -48,8 +48,9 @@
 
   <!-- note that codemeta:affiliation expects Organization as value,
        but https://codemeta.github.io/terms/ mistakenly documents it as Text.
-       See https://github.com/codemeta/codemeta/pull/239 -->
-  <xsd:element name="affiliation" type="codemeta:PersonOrOrganization" />
+       See https://github.com/codemeta/codemeta/pull/239
+       Therefore, more custom checks are implemented in swh/deposit/api/checks.py
+       in order to allow either. -->
 
   <xsd:element name="name" type="xsd:string" />
   <xsd:element name="givenName" type="xsd:string" />