Change second tabset on click in flexdashboard

This can be implemented with JavaScript. Luckily, Knitr supports embedded Javascript. I am an R programmer foremost, so this may not necessarily be the most concise implementation, but it does achieve the correct effect.

    ---
    title: "Untitled"
    output: 
      flexdashboard::flex_dashboard:
        orientation: columns
        vertical_layout: fill
    ---

    ```{r setup, include=FALSE}
    library(flexdashboard)
    ```

    Column {.tabset}
    -----------------------------------------------------------------------

    ### Chart A1
    Chart A1 is Great

    ### Chart B1
    Chart B1 is Great

    Column {.tabset}
    -----------------------------------------------------------------------

    ### Chart A2
    Chart A2 is Great

    ### Chart B2
    Chart B2 is Great

    ```{js}
    // Once the Document Has Fully Loaded
    document.addEventListener("DOMContentLoaded", function(){
      // Select The Tabs
      window.a1Tab = document.querySelector("#column > ul > li:nth-child(1)");
      window.b1Tab = document.querySelector("#column > ul > li:nth-child(2)");
      window.a2Tab = document.querySelector("#column-1 > ul > li:nth-child(1)");
      window.b2Tab = document.querySelector("#column-1 > ul > li:nth-child(2)");

      // Select the Panel Content
      window.a1Panel = document.getElementById('chart-a1');
      window.b1Panel = document.getElementById('chart-b1');
      window.a2Panel = document.getElementById('chart-a2');
      window.b2Panel = document.getElementById('chart-b2');

      // If We Click on B1, Open B2, Close A2
      b1Tab.addEventListener('click', function(){
        a2Tab.classList.remove('active');
        a2Tab.children[0].setAttribute('aria-expanded', true);
        a2Panel.classList.remove('active');

        b2Tab.classList.add('active');
        b2Tab.children[0].setAttribute('aria-expanded', true);
        b2Panel.classList.add('active');
      }, false);

      // If We Click on B2, Open B1, Close A1
      b2Tab.addEventListener('click', function(){
        a1Tab.classList.remove('active');
        a1Tab.children[0].setAttribute('aria-expanded', true);
        a1Panel.classList.remove('active');

        b1Tab.classList.add('active');
        b1Tab.children[0].setAttribute('aria-expanded', true);
        b1Panel.classList.add('active');
      }, false);

      // If We Click on A1, Open A2, Close B2
      a1Tab.addEventListener('click', function(){
        b2Tab.classList.remove('active');
        b2Tab.children[0].setAttribute('aria-expanded', true);
        b2Panel.classList.remove('active');

        a2Tab.classList.add('active');
        a2Tab.children[0].setAttribute('aria-expanded', true);
        a2Panel.classList.add('active');
      }, false);

      // If We Click on A2, Open A1, Close B1
      a2Tab.addEventListener('click', function(){
        b1Tab.classList.remove('active');
        b1Tab.children[0].setAttribute('aria-expanded', true);
        b1Panel.classList.remove('active');

        a1Tab.classList.add('active');
        a1Tab.children[0].setAttribute('aria-expanded', true);
        a1Panel.classList.add('active');
      }, false);
    });

    ```

Edit: For an unlimited number of tabs, assuming that you will always have the same number of tabs in the first and second column. Same disclaimer, this is not necessarily the most concise implementation, but achieves the desired effect.

    ---
    title: "Untitled"
    output: 
      flexdashboard::flex_dashboard:
        orientation: columns
        vertical_layout: fill
    ---

    ```{r setup, include=FALSE}
    library(flexdashboard)
    ```

    Column {.tabset}
    -----------------------------------------------------------------------

    ### Chart A1
    Chart A1 is Great

    ### Chart B1
    Chart B1 is Great

    ### Chart C1
    Chart C1 is Great

    ### Chart D1
    Chart D1 is Great

    Column {.tabset}
    -----------------------------------------------------------------------

    ### Chart A2
    Chart A2 is Great

    ### Chart B2
    Chart B2 is Great

    ### Chart C2
    Chart C2 is Great

    ### Chart D2
    Chart D2 is Great

    ```{js}
    // Once the Document Has Fully Loaded
    document.addEventListener("DOMContentLoaded", function(){
      // Select The Tabs
      window.col1Tabs = document.querySelector("#column > ul");
      window.col2Tabs = document.querySelector("#column-1 > ul");

      // Select the Panel Content
      window.col1Panels = document.querySelector("#column > div");
      window.col2Panels = document.querySelector("#column-1 > div");

      // Function to Make Tabs Active
      window.handleTab = function(tabIndex){
        for(i=0;i<col1Tabs.childElementCount;i++){
          col1Tabs.children[i].classList.remove('active');
          col2Tabs.children[i].classList.remove('active');
          col1Panels.children[i].classList.remove('active');
          col2Panels.children[i].classList.remove('active');
        }
        col1Tabs.children[tabIndex].classList.add('active');
        col2Tabs.children[tabIndex].classList.add('active');
        col1Panels.children[tabIndex].classList.add('active');
        col2Panels.children[tabIndex].classList.add('active');
      }

      // For All Tabs, Add Event Listener
      for(i=0;i<col1Tabs.childElementCount;i++){
        col1Tabs.children[i].setAttribute('onclick', 'handleTab(' + i + ');');
        col2Tabs.children[i].setAttribute('onclick', 'handleTab(' + i + ');');
      }

    });

    ```

Edit: For anyone who may find this question later, a JQuery implementation with more flexibility was added to the Github Issue.

Just to add another answer using JQuery and boostrap tabset JS function. It is more concise than other answer but works the same with any number of tabs (but the same number) in your two column format.

---
title: "Untitled"
output: 
  flexdashboard::flex_dashboard:
    orientation: columns
    vertical_layout: fill
---

```{r setup, include=FALSE}
library(flexdashboard)
```

```{js}
document.addEventListener("DOMContentLoaded", function(){
    $('a[data-toggle="tab"]').on('click', function(e){
      // find the tab index that is click
      child = e.target.parentNode;
      tabnum = Array.from(child.parentNode.children).indexOf(child);
      // find in which column we are
      column = $(e.target).closest("div[id]");
      // show the same tab in the other column
      columnid = column.attr("id");
      if (columnid == "column") {
        columnto = "column-1";
      } else {
        columnto = "column";
      }
      $("div[id="+columnto+"] li:eq("+tabnum+") a").tab('show');
    })
});
```

Column {.tabset}
-----------------------------------------------------------------------

### Chart A1

This is a text

### Chart B1

```{r}
plot(iris)
```


Column {.tabset}
-----------------------------------------------------------------------

### Chart A2

```{r}
plot(mtcars)
```

### Chart B2

This is another text