Reverse proxy with sub-path 🠞 internal redirect loop

I'm setting up a local test instance, but I'm having trouble with one part of the setup. First the basics:

• WordPress is running in a docker container
• nginx is serving as a reverse proxy, providing SSL

This works 100% if I have the proxy path as "/", but I want to put it at "/wp/" so it doesn't conflict with other existing services. When I try to use the sub-path, I get an internal redirect loop:

[Sun Jul 28 15:43:23.855765 2024] [core:debug] [pid 41:tid 41] core.c(3968): [client 172.17.0.1:33772] AH00121: r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855769 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855772 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855775 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855778 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855781 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855784 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855787 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855790 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855793 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/index.php
[Sun Jul 28 15:43:23.855796 2024] [core:debug] [pid 41:tid 41] core.c(3974): [client 172.17.0.1:33772] AH00122: redirected from r->uri = /wp/
172.17.0.1 - - [28/Jul/2024:15:43:23 +0000] "GET /wp/ HTTP/1.1" 500 809 "-" "Mozilla/5.0 (Windows NT 10.0; Win64; x64; rv:128.0) Gecko/20100101 Firefox/128.0"

I though that was the purpose of wp_options.siteurl and wp_options.home. I've tried every combination of "https://<host>/" and "https://<host>/wp/" in those values (where <host> is my actual hostname), but nothing helps.

I've tried a bunch of other solutions I've found on google, but none of them have fixed it.

I feel like this is a WordPress configuration problem, but here's my nginx location for completeness.

    location /wp/ {
            proxy_pass http://localhost:3380;
            proxy_http_version 1.1;

            proxy_set_header X-Forwarded-Proto https;
            proxy_set_header X-Forwarded-Scheme https;
            proxy_set_header Upgrade $http_upgrade;
            proxy_set_header Connection $http_connection;
            proxy_set_header Host $host;
            proxy_cache_bypass $http_upgrade;
    }