Crowds - including swarms of excited schoolchildren waving flags - gathered as they arrived at train station in Runcorn, where they were greeted by David Briggs, the Lord-Lieutenant of Cheshire.
It is the first time the duchess has attended an event with the Queen without her husband Prince Harry.
Mr Curphey, 32, who met the Royals after the performance, said: "The Queen said she thoroughly enjoyed it and it was lovely to see the history of the bridge on the videos".
The Queen has her own train she uses to get around the United Kingdom for special engagements.
The awkward exchange comes as Meghan continues to adjust to life as a full-time royal.
For the occasion, Her Majesty wore an equally headline-grabbing pear-colored coat by Stuart Parvin and a matching hat.
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They then walked through the streets of Chester to the local town hall where they attended a council lunch.
That being said, following her first solo engagement with the Queen in Cheshire, a body language expert says that Markle looked 'a bit bored'.
At least for now, we all have to get used to a pantyhose wearing, nude nail polish rocking, sort of quieter version of Meghan.
The duchess sat next to the queen for the performance.
Meghan's highly anticipated trip on the Royal Train - which is he Queen's private train which ferries her from A to Z for long distance journeys - took place on Wednesday overnight into Thursday. After Meghan heard the answer, she got in and the Queen followed.
Earlier this week, Harry's office announced that the prince and Meghan would travel to Australia, Fiji, Tonga and New Zealand later this year.
WPA Pool / Getty Images The duchess and the queen at the Storyhouse theater in Chester.