Some more info:
I turned on debugging and edited a page (/provost/staff/). Here is the output.
It looks like site_url is incorrect.
16:22:16 /provost/wp-admin/post.php get_current_url_supercache_dir: warning! site_url (https://somesite.com/provost) not found in permalink (https://somesite.com/provost/staff/).
16:22:16 /provost/wp-admin/post.php get_current_url_supercache_dir: Removing SERVER_NAME (somesite.com) and https:// from permalink (https://somesite.com/provost/staff/). Is the url right?
16:22:16 /provost/wp-admin/post.php supercache dir: /home/some/public_html/wp-content/cache/supercache/somesite.comhttps://provost/staff/
16:22:16 /provost/wp-admin/post.php wp_cache_post_id_gc post_id: 16 https://somesite.com/provost/staff/ clearing cache in /home/some/public_html/wp-content/cache/supercache/somesite.comhttps://provost/staff/.
16:22:16 /provost/wp-admin/post.php wp_cache_post_id_gc clearing cache in /home/some/public_html/wp-content/cache/supercache/somesite.comhttps://provost/staff//page/.
16:22:16 /provost/wp-admin/post.php wp_cache_post_id_gc clearing cache in /home/some/public_html/wp-content/cache/supercache/somesite.com/provost//page/.
16:22:16 /provost/wp-admin/post.php Post change: deleting cache files in /home/some/public_html/wp-content/cache/supercache/somesite.com/provost/
16:22:16 /provost/wp-admin/post.php Post change: deleting page_on_front and page_for_posts pages.
We run our backend behind SSL. But, not the front-end. I am not sure where site_url is set, aside from that. (Also, we’re running multisite via subdirectory, and this is happening on all our subsites).