Mesovrika lies next to Vrika, behind the round rise the old maps call Misorachi. Same water, a completely different beach: white pebbles instead of sand, in the water and out of it, and nothing built at all. No sunbeds, no bar, no shade.
That is both its advantage and its drawback. With small children the pebbles are hard work. Without them, it is the quiet version of Vrika.
One thing to know before you go. The large day cruise boats anchor directly in front and their passengers jump in from the deck. When two or three of them gather, they take both the view and the quiet. Early in the morning, or after five in the afternoon, the bay is yours.
In the older accounts the sandy beaches of Antipaxos are Vrika, Mesovrika and Voutoumi, and the pebbled and deserted ones are Rodovani, Skidi and Sarakiniko. Mesovrika sits between the two kinds, which is exactly why people argue about it.
There is no road. The boat is the way in.