Own member event is duplicated in timeline when joining a room

This issue has been migrated from [#9768](https://github.com/matrix-org/synapse/issues/9768).

---

When accepting an invite to a new DM, I consistently (on matrix.org) see the last two events of the timeline being my own member event, duplicated. The request URL for this is https://matrix.org/_matrix/client/r0/sync?since=m1932336751~1.1932336915_757284961_312592_815559230_699804110_2214631_242549737_904922611_187106&timeout=30000&filter=2&_cacheBuster=1532609541718558

```json
{
"events": [
{
"content": {
"displayname": "bruno5",
"membership": "join"
},
"origin_server_ts": 1617890563769,
"sender": "@bruno5:matrix.org",
"state_key": "@bruno5:matrix.org",
"type": "m.room.member",
"unsigned": {
"age": 93068
},
"event_id": "$jcf51330L2Ej9-mcccs6bl5G4fdkB91xKgIkn7vOUe4"
},
{
"content": {
"ban": 50,
"events": {
"m.room.avatar": 50,
"m.room.canonical_alias": 50,
"m.room.encryption": 100,
"m.room.history_visibility": 100,
"m.room.name": 50,
"m.room.power_levels": 100,
"m.room.server_acl": 100,
"m.room.tombstone": 100
},
"events_default": 0,
"invite": 0,
"kick": 50,
"redact": 50,
"state_default": 50,
"users": {
"@bruno4:matrix.org": 100,
"@bruno5:matrix.org": 100
},
"users_default": 0
},
"origin_server_ts": 1617890563855,
"sender": "@bruno5:matrix.org",
"state_key": "",
"type": "m.room.power_levels",
"unsigned": {
"age": 92982
},
"event_id": "$sUotYNlUwJH1r4gskhgpeDzRnmaitJ4K0f1RPh5mobk"
},
{
"content": {
"join_rule": "invite"
},
"origin_server_ts": 1617890563926,
"sender": "@bruno5:matrix.org",
"state_key": "",
"type": "m.room.join_rules",
"unsigned": {
"age": 92911
},
"event_id": "$n5CR2u0GTgCJmpcjKIWegxjyTVKqRytjchxe_sgIsWU"
},
{
"content": {
"history_visibility": "shared"
},
"origin_server_ts": 1617890564015,
"sender": "@bruno5:matrix.org",
"state_key": "",
"type": "m.room.history_visibility",
"unsigned": {
"age": 92822
},
"event_id": "$wFSynF9jBu54-POGzwhsrVS562kHkao6fL6fY2DCEHM"
},
{
"content": {
"guest_access": "can_join"
},
"origin_server_ts": 1617890564101,
"sender": "@bruno5:matrix.org",
"state_key": "",
"type": "m.room.guest_access",
"unsigned": {
"age": 92736
},
"event_id": "$EFiR8kBA8K8CZuKcwSVG7E15TttlJLdCf5ulVaLNF5Q"
},
{
"content": {
"algorithm": "m.megolm.v1.aes-sha2"
},
"origin_server_ts": 1617890564186,
"sender": "@bruno5:matrix.org",
"state_key": "",
"type": "m.room.encryption",
"unsigned": {
"age": 92651
},
"event_id": "$ns1-TrtI8Vm4b7Gtw-rfPTJZc2ZTSPzKtYoJA62_LFA"
},
{
"content": {
"avatar_url": "mxc://matrix.org/AbaQoXqdDHmFzEpEIYBXSTEP",
"displayname": "Bruno the bloodhound",
"is_direct": true,
"membership": "invite"
},
"origin_server_ts": 1617890564287,
"sender": "@bruno5:matrix.org",
"state_key": "@bruno4:matrix.org",
"type": "m.room.member",
"unsigned": {
"age": 92550
},
"event_id": "$oORFPM76LlRaBAnaNxa-K8f_Ut67paUFKcmpz-9o51w"
},
{
"content": {
"algorithm": "m.megolm.v1.aes-sha2",
"ciphertext": "AwgAEoABSOmkHREfvz5cGZqDTvT8cYeTVrTtX+7HQ1xoMH4BCkZlsDLbXXpa1OzzL9bomP8J5EdD9sZVsR7It48CBM3VbIJfyMMBMEq5uSFIkjUyfuZPid57VuZHuNQd93Du8/wxOqUxMar+1rWu3wGXNfT43LlL5TEA2HpTm1JRr01Xu+XF666EE91s4sqZkTPWhtmQVpbSAla1QLrPrKddwN89G/TKXjrb171qGsqdu170U48PIuKBoyBRuVd/XOsoADRFEFlCvSUG8Ac",
"device_id": "NWYSNKCJJL",
"sender_key": "//RkE4avb7fo+VCmbK6HDa7Y4p2sPsMGCyO8/DnAyAU",
"session_id": "JOVjdHpxd34Cm+zp4DVHhSR8UMetkWYB9olMK2rfovA"
},
"origin_server_ts": 1617890576653,
"sender": "@bruno5:matrix.org",
"type": "m.room.encrypted",
"unsigned": {
"age": 80184
},
"event_id": "$ZS3NguTnTHAq0fkIOqutai4Mbfe-MYfQqb1S5tgOK-U"
},
{
"content": {
"avatar_url": "mxc://matrix.org/AbaQoXqdDHmFzEpEIYBXSTEP",
"displayname": "Bruno the bloodhound",
"membership": "join"
},
"origin_server_ts": 1617890656548,
"sender": "@bruno4:matrix.org",
"state_key": "@bruno4:matrix.org",
"type": "m.room.member",
"unsigned": {
"replaces_state": "$oORFPM76LlRaBAnaNxa-K8f_Ut67paUFKcmpz-9o51w",
"prev_content": {
"avatar_url": "mxc://matrix.org/AbaQoXqdDHmFzEpEIYBXSTEP",
"displayname": "Bruno the bloodhound",
"is_direct": true,
"membership": "invite"
},
"prev_sender": "@bruno5:matrix.org",
"age": 289
},
"event_id": "$Qwq0X0QqTvj4ANh0XZtK4okxne4D9E024wz36MtzgF4"
},
{
"content": {
"avatar_url": "mxc://matrix.org/AbaQoXqdDHmFzEpEIYBXSTEP",
"displayname": "Bruno the bloodhound",
"membership": "join"
},
"origin_server_ts": 1617890656548,
"sender": "@bruno4:matrix.org",
"state_key": "@bruno4:matrix.org",
"type": "m.room.member",
"unsigned": {
"replaces_state": "$oORFPM76LlRaBAnaNxa-K8f_Ut67paUFKcmpz-9o51w",
"prev_content": {
"avatar_url": "mxc://matrix.org/AbaQoXqdDHmFzEpEIYBXSTEP",
"displayname": "Bruno the bloodhound",
"is_direct": true,
"membership": "invite"
},
"prev_sender": "@bruno5:matrix.org",
"age": 289
},
"event_id": "$Qwq0X0QqTvj4ANh0XZtK4okxne4D9E024wz36MtzgF4"
}
],
"prev_batch": "t2-1932333599_757284961_312982_815559432_699804383_2214631_242549800_904923309_187106",
"limited": true
}
```