{{-- Site-wide search overlay. Opened by the header search trigger (data-search-open). The form posts to home_url with `s=` + `post_type=product` as a fallback for users without JS — they hit the standard WP search results page. With JS enabled (see app.js, "Search overlay"), typing in the input fires debounced GETs to /wp-json/elit-gastro/v1/search?q=… and renders live results in three groups (products / categories / blog posts). The empty state shows recent searches (from localStorage) and the four homepage product categories as suggestions. Included from layouts/app.blade.php so it's available globally. --}}