How do I add a Quick Look / Add to Cart / Add to Wishlist button?

By including HTML elements as siblings of the $link element. Magento uses some special variables while constructing links like Add to cart and Add to wishlist. You can duplicate the same behavior on Tagalys pages using the following method.

$.fn.tagalys_templates = $.fn.tagalys_templates || {};
$.fn.tagalys_templates.product_tile = function (widget, product_details) {
  var $link = $("<a>", {
    class: "product-link",
    href: product_details.link,
  });
  /* 
    Construct existing product details code
  */
  var tagalysUenc = $.fn.tagalys_platform_variables.uenc;  

  // Create your 'Add to cart' HTML element
  var $add_to_cart_link = $("<button>", {
    class: "add-to-cart non-tagalys",
    type: "button",
    title: "",
    "data-post": JSON.stringify({
      action:
        "/checkout/cart/add/uenc/" +
        tagalysUenc +
        "/product/" +
        product_details.__id,
      data: {
        product: product_details.__id + "",
        uenc: tagalysUenc,
      },
    }),
  }).html($("<span>").html("Add to cart"));  

  // Create your 'Add to Wishlist' HTML element
  var $add_to_wishlist_link = $("<button>", {
    class: "towishlist non-tagalys",
    type: "button",
    title: "Add to wishlist",
    "data-post": JSON.stringify({
      action: "/wishlist/index/add/",
      data: {
        product: product_details.__id + "",
        uenc: tagalysUenc
      },
    }),
  }).html($("<span>").html("Add to Wishlist"));  

  // Append the above created elements as a sibling of $link element.
  return $link.add($add_to_cart_link).add($add_to_wishlist_link);
};