feat(tiles): Tiles header button with a session picker, and a tile's + menu

A Tiles button beside Split in the header, opt-in through the per-device
showTileGridButton setting (read in applyHeaderVisibilitySettings; the
settings checkbox, displayKeys membership and schema exclusion follow with
persistence) and hard-gated like Split: hidden by its --hidden marker, a JS
width check with a live media listener, a @media (max-width: 1179px) backstop
and never in a solo window. While the grid is open the button closes it and
reads as pressed.

Closed, it opens a picker: a checkbox per open session in tab order (never one
popped out to its own window; one with no PTY is offered, its tile shows the
Attach overlay), names as text, preselected with the grid this tab last left,
else the active session and an open split's two. Boxes past what the window
can fit are disabled with the count shown, and Open opens the grid on the
checked sessions, focusing the active one if checked. Escape and an outside
click close it; its close method is idempotent and the global Escape handler
calls it.

A tile's + lists the open sessions not yet tiled; picking one adds it and
focuses it (a human selection). A grid that already holds what the window fits
disables the entries. "New session in this case" waits for auto-join.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

src/web/public/styles.css

@@ -2510,6 +2510,19 @@ html[data-tab-orientation='vertical'].tab-rail-compact
   display: none !important;
 }
 
+/* Tile grid header button: hidden by default (opt-in, `showTileGridButton`),
+   and hard-gated to desktop widths like Split (the backstop below). */
+.btn-tile-grid--hidden {
+  display: none !important;
+}
+
+/* While the grid is open the button's click closes it (aria-pressed too). */
+.btn-icon-header.btn-tile-grid.tiles-open,
+.btn-icon-header.btn-tile-grid.tiles-open:hover {
+  background: var(--accent);
+  color: var(--accent-ink);
+}
+
 /* The tile grid is open (tile-grid.js): the split cannot open beside it. */
 .btn-split.btn-split--blocked {
   opacity: 0.4;
@@ -2537,7 +2550,8 @@ html[data-tab-orientation='vertical'].tab-rail-compact
    HOME_SESSIONS_MIN_WIDTH (home-sessions.js) / SPLIT_PANE_MIN_WIDTH
    (constants.js) — keep the three in sync. */
 @media (max-width: 1179px) {
-  .btn-icon-header.btn-split {
+  .btn-icon-header.btn-split,
+  .btn-icon-header.btn-tile-grid {
     display: none !important;
   }
 }
@@ -2853,6 +2867,7 @@ body.solo-mode .header-tokens,
 body.solo-mode .btn-notifications,
 body.solo-mode .btn-multimonitor,
 body.solo-mode .btn-split,
+body.solo-mode .btn-tile-grid,
 body.solo-mode .header-plan-usage,
 /* A solo window shows ONE session and has no tab strip to put restored ones in,
    so offering to rebuild a list of them there is an offer it cannot show the
@@ -19575,6 +19590,85 @@ html[data-session-list="sidebar"][data-sidebar="collapsed"] .btn-sidebar-toggle
   pointer-events: none;
 }
 
+/* Tiles picker (tile-grid.js openTilePicker): a checkbox per open session,
+   positioned `fixed` under the Tiles button; same layer as the split picker. */
+.tile-picker-menu {
+  position: fixed;
+  z-index: 1000;
+  min-width: 240px;
+  max-width: 340px;
+  max-height: 60vh;
+  display: flex;
+  flex-direction: column;
+  padding: 6px;
+  background: var(--bg-secondary, #1a1a1a);
+  border: 1px solid var(--border-color, #333);
+  border-radius: 6px;
+  box-shadow: 0 8px 24px rgba(0, 0, 0, 0.35);
+  font-size: 13px;
+}
+
+.tile-picker-list {
+  overflow-y: auto;
+  min-height: 0;
+}
+
+.tile-picker-item {
+  display: flex;
+  align-items: center;
+  gap: 8px;
+  padding: 4px 6px;
+  border-radius: 4px;
+  cursor: pointer;
+}
+
+.tile-picker-item:hover {
+  background: color-mix(in srgb, var(--text-muted, #888) 14%, transparent);
+}
+
+.tile-picker-item span {
+  overflow: hidden;
+  white-space: nowrap;
+  text-overflow: ellipsis;
+}
+
+.tile-picker-footer {
+  display: flex;
+  align-items: center;
+  justify-content: space-between;
+  gap: 8px;
+  padding: 6px 4px 2px;
+  border-top: 1px solid var(--border-color, #333);
+  margin-top: 4px;
+}
+
+.tile-picker-hint,
+.tile-picker-empty,
+.tile-add-empty {
+  color: var(--text-muted);
+  font-size: 12px;
+}
+
+.tile-picker-empty,
+.tile-add-empty {
+  padding: 6px 8px;
+}
+
+.tile-picker-open {
+  padding: 4px 12px;
+  font: inherit;
+  color: var(--accent-ink, #fff);
+  background: var(--accent, #4a9eff);
+  border: none;
+  border-radius: 4px;
+  cursor: pointer;
+}
+
+.tile-picker-open:disabled {
+  opacity: 0.5;
+  cursor: default;
+}
+
 /* Split-picker menu: a small popover listing sessions to split with, appended
    to document.body and positioned `fixed` by openSplitPicker() (JS sets
    top/right against the Split header button's own rect). z-index above the