<input type="text" id="searchText" onblur="ShowSelectTag()" />
<input type="button" value="click me" onclick="return false;" />
<select id="SelectToShow" style="width:0px;position:absolute;">
        <option>otpion</option>
        <option>otpion</option>
        <option>otpion</option>
        <option>otpion</option>
        <option>otpion</option>        
    </select>

<script>
   function getPos(el) {
    for (var lx=0, ly=0;
         el != null;
         lx += el.offsetLeft, ly += el.offsetTop, el = el.offsetParent);
    return {x: lx,y: ly};
   }
   (function () {
      var myInput = document.getElementById('searchText');
      var pos = getPos(myInput);
      document.getElementById("SelectToShow").style.left = pos.x+"px";
      document.getElementById("SelectToShow").style.top = pos.y+"px";
   })();

   function ShowSelectTag()
    {
        var Select = document.getElementById("SelectToShow");
        var event;
        event = document.createEvent('MouseEvents');
        event.initMouseEvent('mousedown', true, true, window);
        Select.dispatchEvent(event);
    }
</script>

let me know if it helped !

Try to use a timeout to let the keyboard close before you open the SELECT:

onblur="window.setTimeout('ShowSelectTag()',1000);"

Then try reducing the delay to 0 to see it it works instantly.

onblur="window.setTimeout('ShowSelectTag()',0);"