One evening, after being inspired by https://stackapps.com/q/8500/34061, I did some analysis with [mitmproxy](https://mitmproxy.org/). This was at a time that the publicly released version of the API was 2.2; it turned out that the Stack Exchange mobile apps use a (then) undocumented API v2.3 method for this: [`/me/achievements`](https://api.stackexchange.com/docs/me-achievements).

A `GET` call to `https://api.stackexchange.com/2.3/me/achievements?pageSize=100&filter=*-t3H1VAnvpZbcZ&access_token=[ACCESS_TOKEN]&key=[KEY]` yields the following result:

```json
{
    "quota_max": 10000,
    "quota_remaining": 7999,
    "has_more": false,
    "items": [
        {
            "achievement_type": "reputation",
            "account_id": 6085540,
            "reputation_change": 2,
            "is_unread": false,
            "link": "https://ux.stackexchange.com/questions/130203/whats-the-best-way-to-combine-an-action-outside-of-the-ui-with-an-action-in-the/130208#130208",
            "creation_date": 1574444062,
            "title": "What's the best way to combine an action outside of the UI with an action in the UI?",
            "on_site": {
                "icon_url": "https://cdn.sstatic.net/Sites/ux/img/apple-touch-icon.png",
                "aliases": [
                    "http://ui.stackexchange.com"
                ],
                "site_url": "https://ux.stackexchange.com",
                "api_site_parameter": "ux",
                "name": "User Experience",
                "high_resolution_icon_url": "https://cdn.sstatic.net/Sites/ux/img/[email protected]"
            }
        },
        {
            "achievement_type": "reputation",
            "account_id": 6085540,
            "reputation_change": 10,
            "is_unread": false,
            "link": "https://meta.stackexchange.com/questions/334137/dismiss-the-meet-the-overflow-and-podcast-episode-banners-network-wide/334202#334202",
            "creation_date": 1574443466,
            "title": "Dismiss the "Meet The Overflow" and Podcast Episode banners network-wide",
            "on_site": {
                "icon_url": "https://meta.stackexchange.com/content/Sites/stackexchangemeta/img/apple-touch-icon.png",
                "site_url": "https://meta.stackexchange.com",
                "api_site_parameter": "meta",
                "name": "Meta Stack Exchange",
                "high_resolution_icon_url": "https://meta.stackexchange.com/content/Sites/stackexchangemeta/img/[email protected]"
            }
        },

...
```

It works with any Stack App token, not just the ones from the mobile apps. You'll probably need one with scope `read_inbox`, though I haven't tested that.

There's a related `GET` call to `https://mobile.stackexchange.com/1/push/topbar-counts?filter=vqc7J` which returns the number of unread inbox items and the total new reputation (the green `+x` badge), but it only works with a token which belongs to the mobile apps. It returns something like this:

```json
{
    "total": 1,
    "quota_max": 2147483647,
    "quota_remaining": 2147483647,
    "has_more": false,
    "page": 1,
    "type": "mobile_top_bar_counts",
    "items": [
        {
            "unread_inbox_count": 0,
            "account_id": 6085540,
            "unviewed_non_rep_count": 0,
            "unviewed_rep_count": 0
        }
    ],
    "page_size": 1
}
```

The `filter` seems to be necessary; without it, I get a backoff response.