Do all real-time filters introduce phase lag, or can there be such a thing as "lag-less" or "zero lag" real time filters?
If a filter has any memory and that memory affects the current output, then that past data is lagged into the present.  If that filter is time invariant, then a memory of the present can potentially affect future output. e.g. is lagged into the future.  So, yes, unless the filter is memoryless or not time invariant.  

Note that a unit impulse at the origin is memoryless, but impossible to implement in any finite size system due to the speed of light being finite.