<div class="site">
  <header> 
    <div class="overflow-thing">will overflow</div>
    
    <pre>overflow-x: clip;
overflow-y: visible;</pre>
    
    <div class="popout">Can escape down</div>
  </header>
  <aside>
    <pre>overflow-x: visible;
overflow-y: clip;</pre>
    <div class="overflow-thing">
      will overflow
    </div>
    <div class="popout">
      Something pretty long that needs to escape out to the side. 
    </div>
  </aside>
  <main></main>
</div>

body {
  margin: 0;
  height: 100dvh;
  font: 100%/1.2 system-ui;
}

.site {
  display: grid;
  grid-template:
    "header header"
    "aside main";
  grid-template-columns: 200px 1fr;
  grid-template-rows: 100px 1fr;
  height: 100%;
  
  > * {
    padding: 8px;
  }
}
header {
  background: #999;
  grid-area: header;
  width: 100vw;
  
  overflow-x: clip;
  overflow-y: visible;
  
  .popout {
    inline-size: 100px;
    inset-inline-end: 20px;
    block-size: 100px;
    &::before {
      content: "";
      position: absolute;
      inset-block-end: 100%;
      inset-inline-start: 50px;
      inline-size: 10px;
      block-size: 20px;
      background: linear-gradient(to bottom, #999, white);
    }
  }
  
  .overflow-thing {
    background: red;
    font-size: 9px;
    padding: 2px;
    color: white;
    border-radius: 3px;
    width: 3000px;
  }  
}

aside {
  grid-area: aside;
  position: relative;
  background: #333;
  color: white;
  overflow-y: clip;
  overflow-x: visible;
  height: calc(100dvh - 100px);
  
  .popout {
    inline-size: 300px;
    inset-inline-start: 120px;
    inset-block-start: 200px;
    &::before {
      content: "";
      position: absolute;
      inset-block-start: 20px;
      inset-inline-end: 100%;
      inline-size: 20px;
      block-size: 10px;
      background: linear-gradient(to right, #333, white);
    }
  }
  
  .overflow-thing {
    background: red;
    font-size: 9px;
    padding: 2px;
    color: white;
    border-radius: 3px;
    height: 3000px;
    width: min-content;
  }
}

main {
  grid-area: main;
  background: #555;
}

.popout {
  position: relative;
  background: white;
  color: black;
  position: absolute;
  padding: 8px;
  box-shadow: 0 0 20px lch(0% 0 0 / 0.25);
  border-radius: 4px;
}

* {
  box-sizing: border-box;
}