Очевидно инструментът за избор на дата не позволява на браузъра да задейства събитието за замъгляване, след като е загубил фокус чрез щракване.

Той задейства събитието за замъгляване само ако продължите да движите с табулатори до изгубен фокус или като промените стойностите, като изберете дата.

Така че основно заобиколно решение може да бъде емулиране на събитието замъгляване чрез използване на щракване/фокус на друг елемент.

Заобиколно решение

$(document).ready(function () {
    $(".focus-popover").each(function (index) {
        var showPopover = function () {
            $(this).popover('show');
        };
        var hidePopover = function () {
            $(this).popover('hide');
        };
        $(this).popover({
            html: true,
            placement: $(this).attr('data-placement'),
            trigger: 'manual'
        })
        .focus(showPopover)
        .blur(hidePopover);
    })
    // The nasty Opera < 12 "workaround"
    if ( window.opera && +window.opera.version() <= 13 ) {
        var $buggyInput = $("#test2"), // Caching is important!
            buggyInputFocus = false,
            buggyFocus = function(event) {
                event.stopPropagation();
                if(!buggyInputFocus) {
                    $(this).popover('show');
                    buggyInputFocus = true;
                }
            },
            buggyBlur = function(event) {
                $(this).popover('hide');
                buggyInputFocus = false;
            }
        ;
        // The focus also stop triggering if it was blurred then clicked back, so we added a click. It doesn't run buggyFocus twice since it only execute itself if it hasn't been focused first
        $buggyInput.on({
            "focus": buggyFocus,
            "click": buggyFocus,
            "blur":buggyBlur,
            "change":buggyBlur // On change is also important, you don't want to leave it open when it changes
        })
        // Since it doesn't the blur event, we fake it by capturing focus or click on the html tag
        $("html").on({
            click: function() {
                if ( buggyInputFocus ) $buggyInput.trigger("blur");
            },
            focus: function() {
                if ( buggyInputFocus ) $buggyInput.trigger("blur");
            }
        })
    }
});

Fiddle: http://jsfiddle.net/5wsq38u3/4/

РЕДАКТИРАНЕ: За повече от 1 въвеждане на дата

$(document).ready(function () {
    $(".focus-popover").each(function (index) {
        var showPopover = function () {
            $(this).popover('show');
        };
        var hidePopover = function () {
            $(this).popover('hide');
        };
        $(this).popover({
            html: true,
            placement: $(this).attr('data-placement'),
            trigger: 'manual'
        })
        .focus(showPopover)
        .blur(hidePopover);
    })
    // The nasty Opera < 12 "workaround"
    if (window.opera && +window.opera.version() < 13) {
        var $buggyInputs = $(".operaFix"), // Caching is important!
            buggyInputFocus = false,
            buggyInput = {}, // We store an instance of the focused element
            buggyFocus = function(event) {
                event.stopPropagation();
                if(!buggyInputFocus) {
                    $(buggyInput).popover('hide');
                    $(this).popover('show');
                    buggyInputFocus = true;
                    buggyInput = $(this);
                }
                else if ($(buggyInput).attr("id") !== $(this).attr("id")){
                   $(buggyInput).trigger("blur")
                }
            },
            buggyBlur = function(event) {
                $(this).popover('hide');
                buggyInputFocus = false;
                buggyInput = {}
            }
        ;
        // The focus also stop triggering if it was blurred then clicked back, so we added a click. It doesn't run buggyFocus twice since it only execute itself if it hasn't been focused first
        $buggyInputs.on({
            "focus": buggyFocus,
            "click": buggyFocus,
            "blur": buggyBlur,
            "change": buggyBlur // On change is also important, you don't want to leave it open when it changes
        })
        // Since it doesn't the blur event, we fake it by capturing focus or click on the html tag
        $("html").on({
            click: function() {
                if (buggyInputFocus) $(buggyInput).trigger("blur");
            },
            focus: function() {
                if (buggyInputFocus) $(buggyInput).trigger("blur");
            }
        })
    }
});

JSBin: http://jsbin.com/xucagomutera/1/edit