To solve the optimization problem described, we will use the Gurobi Python interface. The problem involves minimizing an objective function subject to several constraints related to the nutritional content of steaks and sashimi.

The objective function is to minimize `4.37 * steaks + 5.64 * sashimi`, where `steaks` and `sashimi` are the variables representing the quantities of each food item.

Constraints:
1. **Iron from Steaks and Sashimi**: At least 35 milligrams must come from both steaks and sashimi.
2. **Calcium from Steaks and Sashimi**: At least 45 milligrams must come from both steaks and sashimi.
3. **Total Iron Limit**: No more than 165 milligrams of iron can be consumed.
4. **Total Calcium Limit**: No more than 97 milligrams of calcium can be consumed.
5. **Linear Constraint**: `-3 * steaks + 6 * sashimi >= 0`.
6. **Non-Negativity and Integer Constraints**: `steaks` can be any non-negative real number, while `sashimi` must be a non-negative integer.

Given the problem description, we notice some inconsistencies or redundancies in the constraints provided:
- The requirement for at least 35 milligrams of iron from both steaks and sashimi individually might be overly restrictive since it implies that each item alone must meet this threshold. However, the total iron constraint (at least 35 mg from both combined) seems more reasonable.
- Similar reasoning applies to calcium.

We will proceed with interpreting the constraints in a way that makes them feasible for optimization:
- For iron and calcium requirements, we consider the combined minimums rather than individual minimums per item, as this seems more logical given typical dietary considerations.
- We also acknowledge the upper bounds on total iron and calcium intake.

Here is the Gurobi code to solve this problem:

```python
from gurobipy import *

# Create a model
m = Model("Nutrition_Optimization")

# Define variables
steaks = m.addVar(lb=0, name="steaks", vtype=GRB.CONTINUOUS)
sashimi = m.addVar(lb=0, name="sashimi", vtype=GRB.INTEGER)

# Objective function: Minimize 4.37 * steaks + 5.64 * sashimi
m.setObjective(4.37 * steaks + 5.64 * sashimi, GRB.MINIMIZE)

# Constraints
# Iron constraint: At least 35 mg from both
m.addConstr(5 * steaks + 6 * sashimi >= 35, name="iron_minimum")

# Calcium constraint: At least 45 mg from both
m.addConstr(4 * steaks + 10 * sashimi >= 45, name="calcium_minimum")

# Total iron limit: No more than 165 mg
m.addConstr(5 * steaks + 6 * sashimi <= 165, name="iron_maximum")

# Total calcium limit: No more than 97 mg
m.addConstr(4 * steaks + 10 * sashimi <= 97, name="calcium_maximum")

# Linear constraint: -3 * steaks + 6 * sashimi >= 0
m.addConstr(-3 * steaks + 6 * sashimi >= 0, name="linear_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Steaks: {steaks.x}")
    print(f"Sashimi: {sashimi.x}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found")
```

This code should provide the minimum values of `steaks` and `sashimi` that satisfy all given constraints while minimizing the objective function. If no feasible solution exists, it will indicate so based on the optimization status returned by Gurobi.