Campaign groups have received letters reminding them about penalties under new protest laws.
Speaking to the BBC's Today programme, Mr Tugendhat said that anti-monarchy groups have the "liberty that anybody in the United Kingdom has to protest, what they don't have the liberty to do is to disrupt others".
"It's perfectly possible that we're dealing with protest groups that have nothing to do with the UK, but are seeking to protest against a foreign leader who's visiting, or seeking to make a complaint about something that's happening hundreds or thousands of miles away," he said. The Public Order Act includes a penalty of up to six months' imprisonment or an unlimited fine for "locking on" - which involves individuals attaching themselves to others, objects or buildings.
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