As discovered by sunnavy when he went spelunking to determine why
t/walk-web.t started running so many extra tests (7a630ac1), we were
stuffing a bunch of extra query params into the query string,
depending on whether we did that from the hash or passed args (things
like Object and Description).

I rewrote his patch a bit to use the same Field list as search_arguments

We lose saved search state badly in RTIR, but it's not a regression
caused by this code, see [rtir #25241]