Are elections, to choose your leader, allowed in Islam?

- **Are elections, to choose your leader, allowed in Islam?** I have been reading some contradictory opinions on this, on the one hand we have the so called fundamentalists who label it as un-islamic and state that only caliph can rule muslims who is chosen by a group of clerics. On the other hand we have others who say that there is no clear rule of succession in Islam and all previous "Islamic" governments have been infact monarchies and that the act of Election is in effect like giving Bayath to the ruler. - **Is democracy such that the constitution makes it clear that no law can be passed against Islam still shirk?** I read the answers given here https://islam.stackexchange.com/questions/2927/whats-the-limit-of-democracy-in-islam And understood that Democracy in it's most pure form is actually un-islamic. But what is the verdict regarding many "Islamic" countries which argue that since the consitution is based on Islam, the law creating bodies can never pass any law against Islam and is therefore a kind of Islamic democracy.