Overpass Turbo Query - Why does this not work?

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There are three issues to fix. First: Syntax. The minus operator needs parentheses and there is a colon missing.

(
  node["tourism"="camp_site"]["backcountry"="yes"]({{bbox}});
  way["tourism"="camp_site"]["backcountry"="yes"]({{bbox}});
  relation["tourism"="camp_site"]["backcountry"="yes"]({{bbox}});
)->.campsites;

// Find all shelter=lean-to within 50 meters of the campsites in the current map view
(
  node["shelter"="lean-to"](around.campsites:50);
  way["shelter"="lean-to"](around.campsites:50);
  relation["shelter"="lean-to"](around.campsites:50);
)->.nearby_shelters;

// Find all shelter=lean-to in the current map view
( (
  node["shelter"="lean-to"]({{bbox}});
  way["shelter"="lean-to"]({{bbox}});
  relation["shelter"="lean-to"]({{bbox}});
); - .nearby_shelters; );

out body;
>;
out skel qt;

Second: A quick recherche on taginfo shows that it is rather shelter_type than shelter that has a value that is similar to lean-to.

Third, it is rather lean_to than lean_to.

[out:json];

// Find all tourism=camp_site with backcountry=yes in the current map view
(
  node["tourism"="camp_site"]["backcountry"="yes"]({{bbox}});
  way["tourism"="camp_site"]["backcountry"="yes"]({{bbox}});
  relation["tourism"="camp_site"]["backcountry"="yes"]({{bbox}});
)->.campsites;

// Find all shelter=lean-to within 50 meters of the campsites in the current map view
(
  node["shelter_type"="lean_to"](around.campsites:50);
  way["shelter_type"="lean_to"](around.campsites:50);
  relation["shelter_type"="lean_to"](around.campsites:50);
)->.nearby_shelters;

// Find all shelter=lean-to in the current map view
( (
  node["shelter_type"="lean_to"]({{bbox}});
  way["shelter_type"="lean_to"]({{bbox}});
  relation["shelter_type"="lean_to"]({{bbox}});
); - .nearby_shelters; );

out body;
>;
out skel qt;

Finally, let’s clean up the query to get it shorter and faster. Use nwr:

[out:json];

// Find all tourism=camp_site with backcountry=yes in the current map view
nwr["tourism"="camp_site"]["backcountry"="yes"]({{bbox}})->.campsites;

// Find all shelter=lean-to within 50 meters of the campsites in the current map view
nwr["shelter_type"="lean_to"](around.campsites:50)->.nearby_shelters;

// Find all shelter=lean-to in the current map view
( nwr["shelter_type"="lean_to"]({{bbox}}); - .nearby_shelters; );

out body;
>;
out skel qt;

Then, get the query more optimized towards actual data structures:

nwr["shelter_type"="lean_to"]({{bbox}})->.all;
nwr["tourism"="camp_site"]["backcountry"="yes"]({{bbox}})->.campsites;
( .all; - nwr.all(around.campsites:50); );
out center;

This fetches all shelters early and reuses them to compute the ones to skip.

Finally, there is a full manual, so no need for mostly guessing AI ersatz.