I've solved this by adding this RadToolTipManager to my masterpage:
<telerik:RadToolTipManager ID="RadToolTipManager1" runat="server" Position="BottomRight" AutoTooltipify="true" AutoCloseDelay="0" OffsetY="10">
</telerik:RadToolTipManager>
AutoTooltipify="true" converts the standard tooltips to radTooltips. AutoCloseDelay="0" gets me the behaviour that I want.