You're close - just stop the propagation when the user clicks within the popups:
$("#email-popup, #phone-popup").on("click", function(e){
e.stopPropagation();
});
$(".email").on("click", function(e){
e.stopPropagation();
$("#email-popup").show("fast");
});
$(".phone").on("click", function(e){
e.stopPropagation();
$("#phone-popup").show();
});
Also you have some repeated code in the document click. You're hiding the email popup twice.
$(document).on("click", function() {
$("#email-popup, #phone-popup").hide("fast");
});
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